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Serial No. 9 ' 


DEPARTMENT OF COMMERCE 

U. S. COAST AND GEODETIC SURVEY 

E. LESTER JONES, Superintendent 


GEODESY 


APPLICATION OF THE THEORY OF LEAST 
SQUARES TO THE ADJUSTMENT 
OF TRIANGULATION 


BY 

OSCAR S. ADAMS 

COMPUTER 

UNITED STATES COAST AND GEODETIC SURVEY 


Special Publication No. 28 



WASHINGTON 

GOVERNMENT PRINTING OFFICE 
1915 


Monograph 


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Serial No. 9 


2 - 


DEPARTMENT OF COMMERCE 

U. S. COAST AND GEODETIC SURVEY 
*» 

E. LESTER JONES, Superintendent 


GEODESY 


APPLICATION OF THE THEORY OF LEAST 
SQUARES TO THE ADJUSTMENT 

IF* 


OF TRIANGULATION 


BY 

OSCAR S. ADAMS 

* 

COMPUTER 

UNITED STATES COAST AND GEODETIC SURVEY 


Special Publication No. 28 



WASHINGTON 

GOVERNMENT PRINTING OFFICE 
1915 















ADDITIONAL COPIES 

OF THIS PUBLICATION MAY BE PROCURED FROM 
THE SUPERINTENDENT OF DOCUMENTS 
GOVERNMENT PRINTING OFFICE 
WASHINGTON, D. C. 

AT 

25 CENTS PER COPY 

V 


D. of D, 
OCT 26 1915 




va 



* 


CONTENTS. 


Page. 


General statement. 7 

Station adjustment. 7 

Observed angles. 8 

List of directions. 8 

Condition equations. 9 

Formation of normal equations by differentiation. 9 

Correlate equations. 11 

Formation of normal equations. 11 

Normal equations. 12 

Discussion of method of solution of normal equations. 12 

Solution of normal equations. 13 

Back solution. 13 

Computation of corrections. 13 

Adjustment of a quadrilateral. 14 

General statement. 14 

Lists of directions. 16 

Figure. 16 

Angle equations. 17 

Side equation. 17 

Formation of normal equations by differentiation. 17 

Correlate equations. 18 

Normal equations. 18 

Solution of normal equations. 19 

Back solution. 19 

Computation of corrections. 19 

Adjustment of a quadrilateral by the use of two angle and two side equations.. 20 

Angle equations. 20 

Side equations. 20 

Correlate equations.. j_ 20 

Normal equations. 20 

Solution of normal equations. 21 

Back solution. 21 

Computation of corrections. 21 

Solution of a set of normals including terms usually omitted. 22 

Discussion of the solution. 22 

Solution of triangles. 23 

Position computations, secondary triangulation. 24 

List of geographic positions.*. 26 

Development of condition equations for latitude and longitude closures. 26 

Equations in a net. 32 

Adjustment of a figure with latitude, longitude, azimuth, and length closure 

conditions. 34 

Figure. 34 

Angle equations. 35 

Azimuth equation. 35 

Side equations. 35 

Length equation. 37 


3 
















































4 CONTENTS. 

Adjustment of a figure with latitude, longitude, azimuth, and length closure 
conditions—Continued. Page. 

Figure for latitude and longitude equations. 37 

Formation of azimuth equation. 38 

Preliminary computation of triangles. 38 

Preliminary computation of positions, primary form. 40 

Formation of latitude and longitude condition equations. 50 

Latitude equation. 51 

Longitude equation. 51 

Correlate equations. 52 

List of corrections. 55 

Normal equations. 56 

Solution of normals. 58 

Back solution....'. 68 

Computation of corrections. 69 

Final solution of triangles. 71 

Final computation of positions. 76 

List of geographic positions. 91 

Adjustment of triangulation by the method of variation of geographic coordi¬ 
nates. 91 

Development of formulas. 91 

Adjustment of a quadrilateral with two points fixed. 94 

Lists of observed directions. 94 

Preliminary computation of triangles. 95 

Preliminary computation of positions. 96 

Formation of observation equations. 98 

Table for formation of normals, No. 1. 100 

Table for formation of normals, No. 2. 101 

Normal equations. 101 

Solution of normals. 101 

Back solution. 102 

Computation of corrections. 102 

Adjusted computation of triangles. 103 

Adjustment of three new points by variation of geographic coordinates. 103 

General statement. 103 

Figure. 104 

First method. 105 

List of directions. 105 

Lists of fixed positions. 106 

Preliminary computation of triangles. 107 

Preliminary computation of positions. 110 

Formation of observation equations. 114 

Table for formation of normals, No. 1. 118 

Table for formation of normals, No. 2. 119 

Normal equations. 119 

Solution of normals. 120 

Back solution. 121 

Computation of corrections. r ... 121 

Final computation of triangles. 122 

Second method. 125 

Formation of observation equations. 126 

Table for formation of normals, No. 1. 127 

Table for formation of normals, No. 2. 127 

Normal equations. 127 

Solution of normals. 128 






















































CONTENTS. 


5 


Adjustment of three new points by variation of geographic coordinates—Con. 


Second method—Continued. Page. 

Back solution. 129 

Computation of corrections. 129 

Final computation of triangles... 130 

Final computation of positions. 134 

Computation of probable errors. 138 

Adjustment of a figure with latitude and longitude, azimuth, and length condi¬ 
tions by variation of geographic coordinates. 139 

Table of fixed positions.. 139 

Preliminary computation of triangles. 140 

Preliminary computation of positions. 144 

Figure. 157 

Formation of observation equations. 158 

Table for formation of normals, No. 1. 162 

Table for formation of normals, No. 2. 166 

Normal equations. 168 

Solution of normals. 169 

Back solution. ‘ . 174 

Computation of corrections. 175 

Final computation of triangles. 178 

Final computation of positions. 182 

Adjustments by the angle method. 196 

Adjustment of verticals. 197 

General statement. 197 

Figure. 197 

Computation of elevations from reciprocal observations. 198 

Computation of elevations from nonreciprocal observations. 199 

Fixed elevations. 200 

Assumed and adjusted elevations. 200 

Formation of observation equations. 200 

Table of formation of equations. 201 

Computation of probable error. , .. 202 

Formation of normal equations by differentiation. 202 

Table for formation of normal equations. 204 

Normal equations. 204 

Solution of normal equations. 204 

Back solution. 205 

Development of formulas for trigonometric leveling. 205 

General statement. * . 205 

Development of formulas. 207 

Examples of computation by formulas. 214 

Recapitulation of formulas. 216 

Notes on construction and use of tables. 217 

Tables. 218 

Notes on the developments. 219 













































APPLICATION OF THE THEORY OF LEAST SQUARES 
TO THE ADJUSTMENT OF TRIANGULATION 


By Oscar S. Adams 

Computer United States Coast and Geodetic Survey 


GENERAL STATEMENT 

In this publication the aim has not been to develop the theory of 
least squares, but to illustrate the application of the method to the 
problems arising in the adjustment of tri angulation. The general 
idea has been to collect material in one volume that will serve as a 
working manual for the computer in the office and for such other 
members of the Survey as may desire to make these special applica¬ 
tions. It has not been deemed necessary to insert the derivation of 
formulae except in the case of a few special ones that are not usually 
found in the textbooks on least squares. 

For the general theory reference should be made to such books as 
the following: 

Crandall: Geodesy and Least Squares. 

Helmert: Die Ausgleichungsrechnung nach der Metode der kleinsten Quadrate. 

Jordan: Handbuch der Vermessungskunde, volume 1. 

Merriman: Textbook of Least Squares. 

Wright and Hayford: Adjustment of Observations. 

Some of the simpler cases are treated first, such as the local adjust¬ 
ment at a station, the adjustment of a simple quadrilateral, etc. 
After these is given the development of the condition equations for 
latitude and longitude closures, followed by a sample adjustment 
including the condition equations for these closures, together with 
the equations for length and azimuth conditions. 

A method of adjustment by the variation of geographic coordinates 
is then developed and applied first to a quadrilateral, then to a figure 
with a few new points connected with a number of fixed points. 
The same method is applied to the adjustment of a figure with lati¬ 
tude, longitude, length, and azimuth conditions. A sample adjust¬ 
ment of a vertical net is carried through and lastly there is given the 
development of the formulae for the computation of vertical observa¬ 
tions, together with examples of the method of computation. 


7 




8 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

STATION ADJUSTMENT 

i 

The general rule followed by the observers of the Coast and Geo¬ 
detic Survey is to measure the angles at each station in the order of 
azimuth, thus giving rise to no conditions except the horizon closure. 
Occasionally, however, sum angles are observed and, when this is 
done, other conditions are introduced in addition to the horizon 
closure making it necessary to adjust the angles at the station by the 
method of least squares. If all angles were observed in the same 
way, the weight of each would be unity and the adjustment would be 
made without regard to weights. In the adjustment given below the 
angles were measured by the usual Coast and Geodetic Survey 
repetition method; that is, six measures of the angles with the tele¬ 
scope direct and six with it reversed for each set. A station has been 
chosen at which there are angles measured with one, two, and three 
sets in order to illustrate the method of weighting. 


Observed angles , Gray Cliff 



Observed station 

Direction 

Adjusted 

final 

seconds* 


O / // 

n 

Boulder. 

0 00 00.0 

00.0 

Tower. 

65 06 29.3+ri 

29.9 

Tyonek. 

84 52 56. 2+V\+V2 

57.4 

Round Point. 

93 32 12.1+ V \+#3 

13.3 

Moose Point. 

158 04 23. b+Vi+Vi+Vz+Vt 

24.6 

Birch Hill. 

159 55 34. 7+V\+V2+Vz+v$-\-V\q 

35.7 


* These values result from the computation on p. 13. 


< 
































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


9 


There have been formed a complete list of directions without using 
five of the angles, each of which, then, gives rise to a condition, there 
being five conditions in all. The equations expressing these con¬ 
ditions are formed in the following manner: 

Angle Round Point-Boulder, observed, 266 27 47.9+v 4 

Angle Round Point-Boulder, from the list, 266 27 47. 9—v 1 —v 2 —v 3 

Condition No. 1, 0=+0. 0+fl 1 +u 2 +v 3 +tt 4 

Angle Round Point-Bircli Hill, from the list, 66 23 22. 6+v 9 +v 10 

Angle Round Point-Birch Hill, observed, 66 23 21. 8+v 5 

Condition No. 2, 0=+0. 

In the same way the other condition equations are formed. As a 
result there are finally: 

Condition equations 

1. 0=+0. 0+^+^+^+^ 

2. 0=+0. 8—i> 5 +^ 9 +'i; 10 

3. 0=— 3. l+'v 1 +'y 2 + ,y 3+ i; o+ ,y 9 - t“' y io 

4. 0=+0. 0-\-v l -\-v 2 —v 7 

5. 0=+3. 5+v 3 —r 8 -|-'r 9 4-'y 10 

FORMATION OF NORMAL EQUATIONS BY DIFFERENTIATION 

According to the theory of least squares, the most probable values 
will be determined by making the 2 p n v n 2 a minimum, subject to 
the given conditions. By the method of Lagrangian multipliers the 
formation of the normal equations can be much simplified. 

With the use of these the function u that is to be made a minimum 
is 

w=3 V+3 V+3 vf+1 V+2 v*+2 V+l V+l V+2 V+l V-2 

+^4+0.0) — 2C 2 ( — < u 5 + , y 9 + , i; 10 +0.8) — 2C 3 (+'y 1 +r 2 +'*’3+' y 6+'* , 9+ ,l; io — 3.1) — 2C 4 

(+'ih+^2“ ,y 7'h0-0)~2C' 5 (+'V3—'y 8 +' y 9+' u io+3.5). 

The C ’s are merely undetermined multiphers, the values of which 
will be determined by the solution. The factor 2 is included to 
obviate later on the use of the fraction J; the minus sign is used for 
convenience. The function will be rendered a minimum if the 
partial differential coefficients with respect to v 17 v 2 , etc., are equated 
to zero. By this means ten equations will be formed, giving the ten 
v’s expressed in terms of the ^’s. 

Differentiating with respect to v lt v 2 , etc., in succession and equat¬ 
ing the results to zero, the following equations are obtained: 

3 v x — C x —C 3 —C 4 =0 

3 v 2 —C x —C 3 —C 4 =0 

3 v 3 — C l — C 3 — C b —0 
v 4 —*(71=0 

2 + C 2 =0 

2 v 3 — C 3 =0 

V 7 + 04=0 

^8 + C 5 =0 

2 v 9 — C 2 — C 3 — C 5 —0 
^10— 02— 03 05 = ® 


10 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Therefore 

% =+* Ci+J C 3 +$ C 4 
*2 =+i C,+i C 3 +* c 4 

v 3 =+* C,+i C 3 +J C 5 

V 4 =+(?! 

^5 =—^ C 2 

v 6 = + 2 C3 

«7—C4 
^8 = f'S 

^9 = +5 Qj + i Cg + i Cfi 

^io = +Q2+ C3+Q 

Thus all of the v’s are now expressed in terms of the C’s. These 
can now he substituted in the condition equations forming five 
normal equations containing five C’s and these equations may then 
be solved for the C’s. If the normals are formed from these values, 
fractions will occur in practically all of the coefficients. This can 
be avoided by replacing C l by 6 C t ', C 2 by 6 C 2 , etc. This is equiva¬ 
lent to using 12 Ci, 12 C 2 ', etc., in the original function instead of 
2 C v 2 C 2 , etc., which, of course, is perfectly valid. 

The equations will then stand as follows: 

=+2 C/+2 C/+ 2 C' 
v 2 = +2 C/+2 C/+2 C' 

^ =+2 C/+2 C/+2 C 6 ' 
v 4 =+6 C/ 
v 5 =—3 C/ 

^6 C/ 
v 7 ——6 C/ 

v 8 —6 C/ 

v 9 = +3 C/+3 C/+3 C/ 

^io=+6 C/+6 C/+6 C/ 

Dropping the prime and substituting these values in the first con¬ 
dition equation the following normal equation is obtained: 

2 Ci+2 C 3 +2 C 4 +2 Ci-1-2 C 3 +2 C 4 +2 C,+ 2 C 3 +2 <7 5 +6 Ci+0.0=0 
+12 +6 C 3 +4 C 4 +2 C 5 +0.0=0 

In a similar manner the other normal equations are formed, giving 
in all the following five equations: 

+12 C i +6 C 3 + 4 C 4 + 2 C 5 +0.0=0 

+12 C 2 + 9C 3 +9 C 5 +0.8=0 

+ 6 C|+ 9 C 2 +18 C 3 + 4 0 4 +ll C 6 —3.1=0 
+ 4 Ci +4 C 3 +10 0 4 +0.0=0 

+ 2 Ci+ 9 C 2 +ll C 3 +17 C s +3.5=0 

This manner of forming the normal equations is called the method 
of correlates and is most conveniently carried out by means of a 
table of correlates formed as on page 11. 

After the determination of the C’s by the solution of the normal 
equations, the v’s may be computed from the equations of the v’s 


APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


11 


in terms of the C’s. In the tabulated form below the first col¬ 
umn is multiplied by O v the second by C 2 , etc. The sum of the 

first line multiplied by the ^ for that line gives so also for the 
other v’s. 

Correlate equations 



6 

V 

1 

2 

3 

4 

5 

I 

1>’S* 

Adopted 

v’s 

1 

2 

+1 


+1 

+1 


+3 

+0.618 

+0.6 

2 

2 

+ 1 


+ 1 

+ 1 


+3 

+0.618 

+0.6 

3 

2 

+ 1 


+1 


+1 

+3 

-0.050 

-0.0 

4 

6 

+ 1 





+ 1 

-1.182 

-1.2 

5 

3 


-1 




-1 

+0.585 

+0.6 

6 

3 



+1 



+ 1 

+2.133 

+2.1 

7. 

6 




—1 


-1 

+ 1.230 

+1.2 

8 

6 





-1 

-1 

+3. 234 

+3.3 

9 

3 


+1 

+ 1 


+ 1 

+ 3 

-0.069 

-0.1 

10 

6 


+ 1 

+1 


+ 1 

+3 

-0.138 

-0.1 


* These values result from the computation on p. 13. 


FORMATION OF THE NORMAL EQUATIONS 


After the condition equations are tabulated in correlates as above, 
the next step is the formation of the normal equations. In forming 

these the various products must be multiplied by — or by - in which 

p is the weight of the given v and a is some constant. (See the direct 
formation on p. 9.) It is most convenient to choose a so as to make 
most of the values integers, if this can be done without making the 
quantities too large. In this case 6 is the L. C. M. of the p’s, hence 
it is chosen for a. The normal equations are formed by taking the 


algebraic sums of - times the products of the various columns. 
Normal No. 1 is, in symbols— 


• 1 • l+^| • 1 • 2+^| • 1'3+^| ■ 1 • 4+i'| • 1 • 5+,+ (j| • 1 ■ 


The algebraic sum of the sigma products in the formation checks 
or controls the formation of the normals. Each I line in the corre¬ 
lates is the algebraic sum of that line in the table. As is easily seen, 
the sum of the products of this column in the formation of the nor¬ 
mals should check the algebraic sum of the coefficients of the normal. 
On the first normal +12 + 6 + 4 + 2= +24, which is the same as the 
algebraic sum of the products in the correlates. The I column in the 
normals also includes the constant term. In the third normal +6 + 9 
+18 + 4 +11 =+48. In the I columns of the normal +48 — 3.1 
= +44.9. 


1 1] is the constant term of the condition equation. 














12 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Normal equations 



1 

2 

3 

4 

5 

V 

2 

C’s* 

1 

+12 


+ 6 

+ 4 

+ 2 

+0.0 

+24 

-0.19700 

2 


+12 

+ 9 


+ 9 

+0.8 

+30.8 

-0.19531 

3 



+18 

+ 4 

+11 

-3.1 

+44.9 

+0.71069 

4 




+10 


+0.0 

+ 18 

-0. 20547 

5 





+17 

+3.5 

+42.5 

-0.53917 


* These values result from the computation on p. 13. 


DISCUSSION OF METHOD OF SOLUTION OF NORMAL EQUATIONS 

In the normal equations the coefficients in each equation occurring 
before the diagonal term are omitted, as the equations are sym¬ 
metrical with regard to the diagonal line. The set just given when 
written in full is as follows: 



It can be seen that the coefficients may be omitted to the left of 
the diagonal line and each equation may be read from the top down 
to the diagonal term and then across the page. 

The Doolittle method of solution is used. Equation No. 1 is 
copied and then divided by the diagonal term (+12 in this case), the 
signs being changed. Since No. 2 does not occur on No. 1, this also 
is divided at once by the diagonal term with a change of sign. No. 
3 has +6 on No. 1 and +9 on No. 2; accordingly, the divided coeffi¬ 
cients of No. 1 are multiplied by +6 and those of No. 2 by +9 and 
these give the two products on No. 3. These are then added alge¬ 
braically and divided by the diagonal term with change of sign to 
give 0 3 in terms of No. 4 and No. 5 plus a constant term. In a 
similar manner No. 4 and No. 5 are eliminated, the division on No. 5 
giving the value of C-. The back solution is then carried through 
C 4 = +0.17778<7 5 — 0.10962. When the value of C h is substituted, 
<7 4 = —0.09585 — 0.10962= —0.20547. So also for the remaining C’s. 
For an explanation of the omission of the terms before the diagonal 
term, see page 22. For a fqll discussion of the Doolittle method of 
solution, see Wright & Hayford, Adjustment of Observations, page 
114 et seq. 





















APPLICATION OF LEAST SQUARES TO TRIANGULATION 


13 


Solution of normal equations 


1 

2 

3 

4 

5 

*1 

I 

+ 12 


+ 6 

+ 4 

+ 2 

+0.0 

+ 24 

Cx 


- 0.5 

- 0.33333 

- 0.16667 

+0.0 

- 2 


+ 12 

+ 9 


+ 9 

+0.8 

+30.8 


c 2 

- 0.75 


- 0.75 

-0. 06667 

- 2.56667 



+18 

+ 4 

+11 

-3.1 

+44.9 


1 

- 3 

- 2 

- 1 


-12 


2 

- 6.75 


- 6.75 

-0.6 

-23.1 



+ 8.25 

+ 2 

+ 3.25 

-3.7 

+ 9.8 



c 3 

- 0.24242 

- 0.39394 

+0. 44848 

- 1.18788 




+ 10 


+0.0 

+ 18 



1 

- 1.3333 

- 0.6667 


- 8 



3 

- 0.4848 

- 0.7879 

+0. 8969 

- 2.3758 




+ 8.1819 

- 1. 4546 

+0. 8969 

+ 7.6242 




Ci 

+ 0.17778 

-0.10962 

- 0.93184 





+ 17 

+3.5 

+42.5 




1 

- 0.3333 


- 4 




2 

- 6.75 

-0.6 

-23.1 




3 

- 1.2803 

+ 1.4576 

- 3.8606 




4 

- 0.2586 

+0.1595 

+ 1.3554 





+ 8.3778 

+4.5171 

+ 12.8949 





C 6 

-0.53917 

- 1.53917 


Back solution 


5 

4 

3 

2 

1 

-0.53917 

-0.10962 
-0.09585 

+0. 44848 
+0. 21240 
+0.04981 

-0.06667 
+0.40438 
-0.53302 

+0.08986 
+0.06849 
-0.35535 

-0.53917 

-0. 20547 

+0. 71069 

-0.19531 

-0.19700 


Computation of corrections 


1 

2 

3 

4 

5 

-0.197 
+0.711 
-0.205 

-0.197 
+0. 711 
-0.205 

-0.197 
+0. 711 
-0. 539 

-0.197 

+0.195 

-0.197 

6 

-1.182 

+0.195 

3 

+0.585 

+0.309 

2 

+ 0 . 618 

+0.309 

2 

+0.618 

-0.025 

2 

-0. 050 

6 

7 

8 

9 

10 

+0. 711 

+0.205 

+0.539 

-0.195 
+0. 711 
-0.539 

-0.195 
+0. 711 
-0.539 

+0. 711 

3 

+2.133 

+0.205 

6 

+ 1.230 

+0.539 

6 

+3.234 

-0.023 

3 

-0.069 

-0.023 

6 

-0.138 















































14 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

ADJUSTMENT OF A QUADRILATERAL 
GENERAL STATEMENT 

After the local conditions—that is, those arising from the relations 
of the angles to one another at each station—are satisfied there are 
general conditions arising from the geometrical relations necessary 
to form a closed figure which must he satisfied. To illustrate this, 
let the case of a quadrilateral be taken. The angles of each triangle 
should sum up to 180° plus the spherical excess of the triangle. 
Except in rare cases this does not happen with the observed angles; 
therefore condition equations are needed to bring it about. There 
are four triangles in a quadrilateral, but if three of them close the 
other will also close. There will then be three angle equations in 
the quadrilateral. A fourth equation must be included to insure 
that the lines at the pole will pass through the same point. When 
this condition is satisfied, and the triangles are closed, the same values 
will be obtained for the various sides when the computation is carried 
through different triangles. 

In the adjustment of triangulation in the United States Coast and 
Geodetic Survey the method of directions is used; that is, an angle 
is considered as the difference of two directions.* If v t is the correc¬ 
tion to the first direction in order of azimuth at a given station and 
v 2 the correction to the second direction, the correction to the angle 
will be — v t +v 2 , or the algebraic difference of the Us applying to the 
directions. To avoid the use of so many Us, the custom is to write 
(1) instead of thus the angle given above will have the correction 
symbol — (1) + (2), in which 1 and 2 are not quantities but the sub¬ 
scripts of the corresponding Us. 

An angle equation simply states that the sum of the corrections 
to the angles of a given triangle is to equal the failure in the closure 
of the triangle. In the triangle A 3 A 2 A t (see fig. 1 on p. 16) the angle 
at A 2 is to be corrected by — (1) + (2), the angle at A x by — (4) + (6), 
and the angle at A 3 by — (8) + (9). The sum of the angles needs to 
be increased by 2".3 to make up the sum of 180° plus the spherical 
excess. (See triangle on p. 23.) Therefore — (1) + (2) — (4) + (6) 

— (8) + (9) = + 2.3, or, as it is usually written, 0 = — 2.3 — (1) + (2) 

— (4) + (6) — (8) + (9). 

Three angle equations in a quadrilateral will bring about the 
closure of the four triangles, but it is possible to have all of the tri¬ 
angles close and still the sides fail to check when computed through 
different triangles. To make the computation of lengths consistent 
a side equation must be added to the three angle equations. In 
figure 1 on page 16 the sides can be made consistent in the following 
manner: 


* See lists of directions on p. 16. 



APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


15 


In the triangle A t A 4 A 2 

side J. 2 .4 4 _ sine angle A r sine [—(5)+(6)] 
side ^hvl 4 ~sine angle + 2 ~sine [—(l)+(3)] 

in the triangle A t A 3 A 4 

side A+^sine angle + 3 _ sine [—(7)+(9)] 
side A 3 A 4 sine angle A~ sine [—(4)+(5)] 

and in the triangle A 2 A 3 A 4 

side A 3 A 4 _ sine angle + 2 _ sine [—(2)+(3)] 

side ^4 2 + 4 “sine angle A ~sine [—(7)+(8)j 

If the sides are consistent, the product of these three equations 
gives 

sine [—(5)+(6)l sine [-(7)+(9)l sine [-(2)+(3)] _ 

sine [—(l)+(3)] sine [-(4)+(5)] sine [-(7)+(8)] _i 

In a spherical triangle the same equation is obtained by using the 
sine of the side in place of the side. In the end the equation given 
above results, since the sines of the sides cancel out as did the sides 
above. 

Passing to logarithms, we have 

log sine [-(5)+(6)]+log sine [-(7)+(9)]+log sine [-(2)+(3)]—log sine [—(l)-f-(3)] 
-log sine [—(4)-f-(5)]—log sine [-(7)+(8)]=0 

As this will not be exactly true when the observed angles or angles 
adjusted only for closing errors of the triangles are used except in 
rare cases, a condition equation must be formed to accomplish this 
result. From the table of logarithms we find the amount of change 
of the log sine of the given angle for 1" change in the angle, and this 
multiplied by the v’s applying to the angle will give the change in 
the log sine of that angle. It is customary to consider the log sines 
in six places of decimals, hence the change in the log sine for 1" 
will be taken as units in the sixth place of decimals. 

Of// 

log sine [-(5)+(6)] 20 50 56. 7=9. 5513374 - 5. 53(5)+5. 53(6) 
log sine [—(7)+(9)] 61 47 35.0=9.9450972-1.13(7)-f-1.13(9) 
log sine [-(2)+(3)] 32 09 01. 2=9. 7260280-3. 35(2)+3. 35(3) 

Total.=9. 2224626-3. 35(2)+3. 35(3)-5. 53(5)+5. 53(6) 

-1.13(7)+1.13(9) 

o / // 

log sine [-(l)+(3)] 133 53 46. 3=9. 8576926+2.03(1)-2. 03(3) 
log sine [—(4)+(5)] 26 40 23. 5=9. 6521506 -4.19(4)+4.19(5) 

log sine [—(7)+(8)] 31 03 42. 5=9. 7126180-3. 50(7)+3. 50(8) 


Total.=9. 2224612+2. 03 (1) -2.03(3)-4.19 (4)+4.19 (5) 

-3. 50(7)+3. 50(8) 

9. 2224626 -3. 35(2)+3. 35(3) -5. 53(5)+5. 53(6)-1.13(7) 

+1.13(9) 

9. 2224612+2. 03(1) -2. 03(3) -4.19(4)+4.19(5) -3. 50(7)+3. 50(8) 

0=+l. 4-2. 03(1)—3. 35(2)+5. 38(3)+4.19(4)-9. 72(5)+5. 53(6)+2. 37(7)-3. 50(8) 

+1.13(9) 


91865°—U 


-2 















16 COAST AND CEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


(See tabulated form of this equation on p. 17.) 

This condition requires the lines from A 2 , A v and A 3 to. pass through 
the same point at A 4 . If Us are found that satisfy this equation, at 
the same time satisfying the three angle equations given on page 17, 
they will render the quadrilateral consistent in all respects. 

In a full quadrilateral (see figure 1) there are four conditions. 
These can be put in as three angle equations and one side equation, 
or two angle and two side, or one angle and three side equations. 
(See article by C. A. Schott, Appendix No. 17, United States Coast 
and Geodetic Survey Report of 1875, p. 280.) To illustrate the fact, 
a quadrilateral is adjusted using two angle equations and two side 
equations. (See p. 20.) In order to hold the closure of the triangles, 
the logarithms in the side equations must be found at least to seven 
places to hold the closure to tenths of seconds. Of course this 
method would never be used in practice, as the side equations require 
much more work, but the fact is interesting as an illustration of what 
can be done in the method of adjustment. Four side equations or 
four angle equations 
could not be used, for / 4 

the fourth is function¬ 
ally related to the 
other three, and hence 
they would not be inde¬ 
pendent conditions. 

In a set of equations, 
if an identical one is 
included, the diagonal 
term of the reduced normal will become zero with the possible excep¬ 
tion, of course, of a few units in the last place of solution due to 
accumulations. In any case, if the reduced diagonal term falls below 
unity, there may be danger of instability, since in this case any 
accumulations in the last place of the solution are increased when the 
normal is divided by this term. 



Fig. 1. 


Lists of directions 


Stations observed 

Directions 
after local 
adjustment 

Final 

sec¬ 

onds* 

Stations observed 

Directions 
after local 
adjustment 

Final 

sec¬ 

onds* 

Station A\ 

0 in 


Station A 3 

or tr 


A 3 . 

0 00 00.0 

59. 8 

Ai 

0 00 00.0 

no 7 

A\ . 

26 40 23.5 

23.5 

A 2 _ 

31 03 42! 5 

vU. 4 

42. 0 

A» . 

• 47 31 20.2 

20.5 

Ai... 

61 47 35.0 

34! 8 

Station A 2 



Station A 4 

A\ . 

0 00 00.0 

59.5 

A 2 «• -« 

0 00 00.0 

on 1 

A 3 . 

101 44 45.1 

46.1 

Ai 

25 15 16.2 

uu. 1 

16 Q 

Ai . 

133 53 46.3 

45.8 

A 3 

116 47 20! 0 

IU. t7 

19.2 





* These values result from the following computation. 
























APPLICATION OF LEAST SQUARES TO TRIANGULATION. 17 


Angle equations * • 

0=-2.3-(l)+(2)-(4)+(6)- (8)+ (9). 
0=+3.6-(2)+(3)-(7)+(8)-(10)+(12). 
0=+2.2-(4)+(5)-(7)+(9)-(ll)+(12). 


Side equation 


Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

-7+9 

-5+6 

-2+3 

61 47 35.0 
20 50 56. 7 
32 09 01. 2 

9.9450972 
9.5513374 
9. 7260280 

+1.13 
+5.53 
+3.35 

-4+5 

-1+3 

-7+8 

26 40 23.5 
133 53 46. 3 
31 03 42. 5 

9.6521506 
9. 8576926 
9. 7126180 

+4.19 
-2.03 
+3.50 

9. 2224626 

9. 2224612 


0=+1.4 -2.03(1) -3.35(2)+5.38(3)+4.19(4) -9.72(5)+5.53(6)+2.37(7) -3.50(8)+ 

1.13(9). 


FORMATION OF NORMAL EQUATIONS BY DIFFERENTIATION 

The function u to be rendered a minimum is the sum of the squares 
of the v’q, subject to the four given conditions. 

u=v 1 *+v 2 2 +v 3 a +v 4 2 +v fi 2 +Vi 2 +v 7 a +v 8 2 +Vg 2 +v l0 2 +v n 2 +v l2 2 -2C l (-2.3-v l +v 2 -v 4 + 
v 6 -v 3 +Vg)-2C 2 (+3£-v 2 +v 3 -v 7 +v 3 -Vi 0 +'Vi 2 )-2C 3 (+2.2-v 4 +v 3 -v 7 +Vg-v n 
+v 12 )—2 C 4 (+1.4—2.03 3.35 ^+5.38^+4.19 v 4 -9.72v 5 +5.53 v 6 +2.37i; 7 -3.50 

^8+1*13 v g ) 

Differentiating with respect to the v’s in succession and equating 
to zero, there result after transposition the following equations: 

Vl = — Cj—2.03 C 4 
v 2 =+C 1 -C' 2 -3.35 C 4 
v 3 =+C 2 +5.38 C 4 
v 4 ——C x — Cg+4.19 C 4 
v B -+Ct-9.72 C 4 
^6=+^i+5.53 C 4 
v 7 =-C 2 -C 3 + 2.37 C 4 
^-Ci+Ca-3.50 C 4 
% =+C' 1 +C 3 +1.13 C 4 
^10= — C 2 
v n = — C 3 
^12=+C f 2+C3 

By the substitution of these values in the four condition equations 
the following normal equations result: 

+6 Ci—2 C 2 +2 C 3 +4.65 C 4 -2.3=0 
-2 Ci+6 Cj+2 C3+2.86 C* 4 +3.6=0 
+2 C x +2 C 2 +6 Ca-15.15 C 4 +2.2=0 
+4.65 Cj+2.86 C 2 —15.15 C 3 +206.0470 C 4 +1.4=0 

These normal equations are formed most easily by means of the 
tabular form of the correlate equations given on page 18 . 


* For triangles see p. 23. 

















18 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


• The sum of the squares of each column gives the diagonal term 
in that equation in the normals. All coefficients before the diagonal 
term are omitted; each equation is read by starting at the top of 
the tabular form below, reading down the column to the diagonal 
term, and then along the horizontal line. Compare the full nor¬ 
mals given above with the tabular form below. After the diag¬ 
onal terms are determined column No. 1 in the correlates is multi¬ 
plied by column No. 2 and the algebraic sum of the products taken 
for the coefficient of normal No. 1 on No. 2; this is also the coefficient 
of No. 2 on No. 1. Column No. 1 times No. 3, with the algebraic 
sum of the products, gives the coefficient of No. 1 on No. 3 in the 
normals; also No. 3 on No. 1. Finally, the algebraic sum of the 
products of column No. 1 by column No. 4 gives the coefficient of 
normal No. 1 on No. 4. The algebraic sum of the products of col¬ 
umn No. 1 by the 2 column should check the algebraic sum of the 
coefficients of normal No. 1. To this should be added algebraically 
the constant term of normal No. 1 and the sum placed in the 2 col¬ 
umn of normal No. 1. (See the table of normals below.) 

In the same way the sum of the products of col umn No. 2 times 
column No. 3 is determined for the second normal, and by continuing 
the process all of the normals are formed. 

After the G ’s are determined by the solution of the normals the v’s 
are most conveniently computed by multiplying column No. 1 in 
the correlates by G 1 , column No. 2 by C 2 , col umn No. 3 by C 3 , and 
column No. 4 by <7 4 . Then the algebraic sum of line No. 1 gives v x ; 
of No. 2, v 2 , etc. (See the computation of the v’s on p. 19.) 


Correlate equations 



1 

2 

3 

4 

2 

v’s* 

Adopted 

v’s 


1 

-1 



-2.03 

-3.03 

-0.503 

-0.5 

0.25 

2 

+1 

— 1 


-3.35 

-3.35 

+1.004 

+1.0 

1.00 

3 


+1 


+5.38 

+6.38 

-0.501 

-0.5 

0.25 

4 

-1 


-1 

+4.19 

+2.19 

-0.227 

-0.2 

0.04 

5 



+1 

-9.72 

-8.72 

-0.015 

-0.0 

0.00 

6 

+1 



+5.53 

+6.53 

+0.242 

+0.3 

0.09 

7 


-1 

-1 

+2.37 

+0.37 

+0.663 

+0.7 

0. 49 

8 

-1 

+1 


-3.50 

-3.50 

-0. 493 

-0.5 

0.25 

9 

+1 


+1 

+1.13 

+3.13 

-0.170 

-0.2 

0.04 

10 


-1 



—1 

+0.099 

+0.1 

0.01 

11 



-1 


-1 

+0. 740 

+0.7 

0. 49 

12 


+1 

+1 


+2 

-0.840 

-0.8 

0.64 








2 t?2 

3.55 


Normal equations 


1 

2 

3 

4 

5 

2 

C’s* 

+6 

-2 

+2 

+ 4.65 

-2.3 

+ 8.35 

+0.6547 


+6 

+2 

+ 2.86 

+3.6 

+ 12.46 

-0. 0994 



+6 

- 15.15 

+2.2 

- 2.95 

-0. 7401 




+206.0470 

+1.4 

+199. 8070 

-0.07461 


* These values result from the computation on p. 19. 





















APPLICATION OF LEAST SQUARES TO TRIANGULATIOX 


19 


Probable error of an observed direction = = 


Solution of normal equations 


0.6745 


/ 3.55 
\~4T 


= 0 . 6 . 


1 1 

2 

3 

4 

? 

t 

+6 

-2 

+2 

+ 4.65 

-2.3 

+ 8.35 

Ci 

+0.33333 

-CL 33333 

- a 775 

+0.3S333 

- L 39167 


+6 

+2 

+ 2.S8 

+3.6 

+ 12.46 

1 

—a 6667 

• +a 6667 

+ L55 

-d 7667 

+ 2.7S33 


+5.3333 

+2.6667 

+ 4.41 

-*-2. $333 

+ 15.2433 


c. 

[ —a aoooi 

- 0.S36SS 

—0.53125 

- 2LS5S14 



1 +6 

- 15.15 

+2L 2 

- 2195 


1 

—0.6667 

- Loo 

+0.7667 

- 2.7833 


2 

-1.3333 

- 2.305 

—L 4167 

- 7.6217 




- IS. 905 

+L55 

— 13.355 



c* 

+ 4.72625 

—a 3875 

+ S. 33s o 




+306.0470 

+L4 

+ 199. SOTO 



1 

- 3.6038 

+1.7825 

- 6.4712 



2 1 

- 3.6465 

-2.342$ 

- 12L6044 



3 

- 99.349S 

+7.3257 

- 63.1191 




+109.4469 

+S.1654 

+117.6123 




c« 

-0107461 

— L 07461 


Bach solution 


4 

3 

2 

1 

—a 07461 

-0L3573 
“0.352b 

-0.5312 

+0.0617 

+0.3701 

+0.3533 
+O.057S 
+0.2467 
-a 0331 

-0107461 

-a 7401 

-0.0994 

+0.6547 


Computation of corrections 




1 

2 

3 

4 

5 

6 

-OL 6547 
+0L1515 

+0.6547 
+0.0094 
+0.2499 

-a 0994 
-a 4014 

—0.6547 
+0.7401 
-a 3126 

-0.7401 

+ar252 

+0.6547 
-a 4126 

-0.5032 
— 0.5 

-a 5008 
-Hi 5 

-a 0149 
—0.0 

-rQ- 2421 
+0.3 

+ 1.0040 
+L0 

-A 2272 
-0.2 


S 

9 

10 

11 

12 

+010994 
+0.7401 
—fll life 

—0.6547 
-410994 
-0.2611 

+0.6547 
— Q. 7401 
—0.0843 

+0.0994 

+0.7401 

-0.0994 
-a 7401 

+O.CI994 

+ai 

+0.7401 

+ai 

-O.S395 

-as 

+0.6627 

+0.7 

-0.4930 
—Gl o 

-a 1697 
-0.2 


































































20 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


ADJUSTMENT OF A QUADRILATERAL BY THE USE OF TWO ANGLE 

AND TWO SIDE EQUATIONS * 

(See fig. 1 on p. 16.) 

Angle equations 

0=-2.3-(l)+(2)-(4)+(6)-(8)+(9) 

0=+S.6-(2)+(3)-(7)+(8)-(10)+(12) 

Side equations 


Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

Symbol 

• 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

-7+9 

-5+6 

-2+3 

61 47 35.0 

20 50 56.7 

32 09 01.2 

9. 9450972 

9. 5513374 

9. 7260280 

+1.13 
+5.53 
+3.35 

-4+5 

-1+3 

-7+8 

26 40 23.5 
133 53 46.3 

31 03 42.5 

9.6521506 

9.8576926 

9. 7126180 

+4.19 
-2.03 
+3.50 

9. 2224626 

9. 2224612 


0=+l.4—2.03(1) —3.35(2)+5.38(3)+4.19(4) —9.72(5)+5.53(6)+2.37(7) -3.50(8) 

+1.13(9) 


-2+3 

32 09 01.2 

9. 7260280 

+3. 35 

-7+8 

31 03 42.5 

9. 7126180 

+3.50 

-11 + 12 

91 32 03.8 

9.9998442 

-0. 06 

-4+5 

26 40 23.5 

9. 6521506 

+4.19 

-8+9 

30 43 52.5 

9. 7084309 

+3. 54 

-1+2 

101 44 45.1 

9. 9908094 

-0. 44 

-5+6 

20 50 56.7 

9. 5513374 

+5.53 

-10+11 

25 15 16.2 

9. 6300613 

+ 4. 46 



8.9856405 




8.9856393 



0=+1.2—0.44(1)—2.91(2)+3.35(3)+4.19(4)—9.72(5)+5.53(6)+3.50(7)—7.04(8) 
+3.54(9)+4.46(10) -4.40(11) -0.06(12) 

Correlate equations 



1 

2 

3 

4 

I 

fl'sf 

Adopted 

v’s 

1 

-1 


-2.03 

-0. 44 

- 3.47 

-0. 495 

-0.5 

2 

+ 1 

-1 

-3.35 

-2. 91 

- 6.26 

+0. 996 

+ 1.0 

3 


+ 1 

+5.38 

+3.35 

+ 9.73 

-0. 502 

-0.5 

4 

-1 


+4.19 

+4.19 

+ 7.38 

-0. 227 

-0.2 

5 



-9. 72 

-9. 72 

-19. 44 

-0.013 

-0.0 

6 

+1 


+5. 53 

+5. 53 

+ 12.06 

+0. 240 

+0.3 

7 


-1 

+2.37 

+3. 50 

+ 4.87 

+0. 659 

+0.7 

8 

-1 

s +1 

-3.50 

-7. 04 

-10. 54 

-0.500 

-0.5 

9 

+1 


+1.13 

+3. 54 

+ 5.67 

-0.159 

-0.2 

10 


-1 


+4.46 

+ 3.46 

+0.113 

+0.1 

11 




-4. 40 

- 4.40 

+0. 717 

+0.7 

12 


+ 1 


-0.06 

+ 0.94 

-0.830 

-0.8 


Normal equations 



1 

2 

3 

4 

V 

2 

C's* 

1 

+6 

-2 

+ 4.65 

+ 9.45 

-2.3 

+ 15.80 . 

+0. 2328 

2 


+ 6 

+ 2.86 

- 8.80 

+3.6 

+ 1.66 

-0.8398 

3 



+206.0470 

+208. 2153 

+ 1.4 

+423.1723 

+0.16435 

4 




+276. 0980 

+ 1.2 

+486.1633 

-0.16302 


* For triangles, see p. 23 

f These values result from the computation on p. 21. 

















































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


21 


Solution of normal equations 


1 

2 

3 

4 


I 

+6 

-2 

+ 

4.65 

+ 9.45 

-2.3 

+ 15.80 

Ci 

+0.33333 

— 

0. 775 

- 1.575 

+0.38333 

- 2.63333 


+6 

+ 

2.86 

- 8.80 

+3.6 

+ 1.66 

1 

-0.6667 

+ 

1.55 

+ 3.15 

-0. 7667 

+ 5.2667 


+5.3333 

+ 

4.41 

- 5.65 

*+2.8333 

+ 6.9266 


c 2 

— 

0 . 82688 

+ 1.05938 

-0.53125 

1.29S75 



+206.0470 

+208. 2153 

+ 1. 4 

+423.1723 


1 

— 

3. 6038 

- 7.3238 

+ 1.7825 

- 12.2450 


2 

— 

3. 6465 

+ 4.6719 

-2.3428 

- 5.7275 



+ 198.7967 

+205. 5634 

+0. 8397 

+405.1998 




c 3 

- 1.034038 

-0.004224 

- 2.038262 





+276.0980 

+ 1.2 

+486.1633 




1 

- 14.8838 

+3. 6225 

- 24.8850 




2 

- 5.9855 

+3.0015 

+ 7.3379 




3 

-212. 5604 

-0.8683 

-418.9920 





+ 42.6683 

+6.9557 

+ 49.6240 





Ci 

-0.16302 

- 1.16302 


Back solution 


4 

3 

2 

1 

-0.16302 

-0.00422 
+0.16857 

-0. 5312 
-0.1727 
-0.1359 

+0.3833 
+0. 2568 
-0.1274 
-0. 2799 

-0.16302 

+0.16435 

-0. 8398 

+0.2328 


Computation of corrections 


1 

2 

3 

4 

5 

6 

-0. 2328 
-0.3336 
+0. 0717 

+0. 2328 
+0. 8398 
-0. 5506 
+0. 4744 

+0.9964 
+ 1.0 

-0. 8398 
+0.8842 
-0. 5461 

-0. 2328 
+0.6886 
-0. 6831 

-1.5975 
+ 1.5846 

-0.0129 
-0.0 

+0. 2328 
+0. 9089 
-0.9015 

-0. 4947 
-0.5 

-0.5017 
-0.5 

-0. 2273 
-0.2 

+0. 2403 
+0.3 

7 

8 

9 

10 

11 

12 

+0. 8398 
+0.3895 
-0. 5706 

-0. 2328 
-0. 8398 
-0. 5752 
+1.1477 

—0.5001 

-0.5 

+0. 2328 
+0.1857 
-0. 5771 

+0. 8398 
-0. 7271 

+0. 7173 

+0. 7173 
+0.7 

-0. 8398 
+0.0098 

+0.1127 
+0.1 

-0. 8300 
-0.8 

+0. 6587 
+0.7 

-0.1586 
-0.2 



















































22 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28 . 


SOLUTION OF A SET OF NORMALS INCLUDING TERMS USUALLY 

OMITTED 

A set of four normal equations is solved below with inclusion of the 
terms omitted in the Doolittle method of solution. 


Solution of normals 


1 

2 

3 

4 

i? 

1 







+6 

-2 

+ 2 

+ 4.65 

-2.3 

+ 8.35 

-1 Ci 

+0.33333 

- 0.33333 

- 0.775 

+0.38333 

- 1.39167 

-2 

+6 

+ 2 

+ 2.86 

+3.6 

+ 12.46 

+2 (1) 

-0.6667 

+ 0.6667 

+ 1.55 

-0.7667 

+ 2.7833 


+5.3333 

+ 2.6667 

+ 4.41 

+2.8333 

+ 15.2433 


-1 C 2 

- 0.50001 

- 0.82688 

-0.53125 

- 2.85814 

+2 

+2 

+ 6 

- 15.15 

+2.2 

- 2.95 

-2 

+0.6667 (1) 

- 0.6667 

- 1.55 

+0.7667 

- 2.7833 


-2.6667 (2) 

- 1.3333 

- 2.205 

-1.4167 

- 7.6217 



+ 4 

- 18.905 

+1.55 

- 13.355 



- 1 C 3 

+ 4.72625 

-0.3875 

+ 3.33875 

+4.65 

+2.86 

-15.15 

+206.0470 

+1.4 

+199.8070 

-4.65 

+1.55 

- 1.55 (1) 

- 3.6038 

+1.7825 

- 6.4712 


-4.41 

- 2.205 (2) 

- 3.6465 

-2.3428 

- 12.6044 



+18.905 (3) 

- 89.3498 

+7.3257 

- 63.1191 




+109.4469 

+8.1654 

+117.6123 




- 1 C 4 

-0.07461 

- 1.07461 


DISCUSSION OF THE SOLUTION 

The quantities in heavy type are the ones omitted in the Doolittle 
method of solution of normal equations. They sum up to zero with 
the possible variation of a few units in the last place of the solution. 
This shows that the method is one of curtailed substitution. It can 
also be seen that the quantity in the 2 column is the direct sum of all 
the quantities in each horizontal line including those in heavy type. 
All of tbe quantities in heavy type occur in the regular solution. 
This is o value in the control of the solution. If an equation fails to 
check the 2 column after it is added up, the error can generally be 
located by adding back through noting that the coefficient is changed 
in sign because it is multiplied by — 1. Note the product of equation 
No. 1 on No. 4; —1.55 and +1.55 are the products of No. 1 on No. 3 
and No. 2, respectively; —4.65 is the coefficient of No. 4 on No. 1 
with sign changed. The method is the same in all cases. Care should 
be taken with such coefficients as No. 2 and No. 3 on No. 1. They have 
the same value with opposite sign. If a mistake should be made on 
them the 2 column control would not catch it. Care should be taken 
not to make a mistake in the rj column and a compensating one in the 
2 column. There is most danger of this in the addition. The control 
would not catch this and it would take much labor to correct it later. 












APPLICATION OF LEAST SQLTARES TO TRIANGITLATION. 23 

After each equation is added, it should be added horizontally to 
check the I column. If the check fails an error has been made and 
it must be found before proceeding. A slight variation in the last 
place of the solution is of course unavoidable. After the division 
of each equation by the reduced diagonal term, a horizontal addition 
should be made (including, of course, —1) to check the correctness 
of the division. No time is ever lost in using care in the solution of 
the equations. It takes so much time and labor tp rectify a mistake 
later that every means should be employed to detect and correct it 
in the solution. The larger the set, the more important it is to be on 
guard against errors. It is possible to carry a set through with almost 
absolute assurance that the solution is correct. 

If, in a given equation, the solution fails to check and the check of 
adding back through is satisfied, a mistake has been made somewhere 
in the solution columns and a compensating mistake in the I column. 
This can be caught by building up the omitted columns to the left of 
the given equation. They should each sum up to zero. If any one 
does not, the mistake in addition has been made in that equation in 
the column of the one being eliminated. 


Solution of triangles * 


Symbol 

Station 

Observed angle 

Correction 

Spheri¬ 

cal 

angle 

Spherical 

excess 

Plane 

angle 

Logarithm 


As—A i 

O 

, 

// 


ft 

„ 

ft 

3.772745 

-8+9 

A$ 

30 

43 

52.5 

+0.3 

52.8 

0.0 

52.8 

0.291568 

-1+2 

As 

101 

44 

45.1 

+ 1.5 

46.6 

0.1 

46.5 

9.990809 

-4+6 

Ai 

47 

31 

20.2 

+0.5 

20.7 

0.0 

20.7 

9.867787 






+2.3 


0.1 




A z -A x 








4.055122 


A 3 —As 








3.932100 


As-Ai 








3. 772745 

-10+11 

At 

25 

15 

16.2 

+0.6 

16.8 

0.0 

16.8 

0*169936 

-1+3 

As 

133 

53 

46.3 

+0.0 

46.3 

0.1 

46.2 

9 -£57693 

-5+6 

Ax 

20 

50 

56.7 

+0.3 

57.0 

0.0 

57.0 

8. 551339 






+0.9 


0.1 




Ax-Ax 








4.000374 


Ax-As 








3.694020 


As—A3 








3.932100 

-10+12 

Ax 

116 

47 

20.0 

-0.9 

19.1 

0.1 

19.0 

0.049306 

-2+3 

As 

'32 

08 

61.2 

-1.5 

59.7 

0.0 

59.7 

9. 726023 

-7+8 

As 

31 

03 

42.5 

-1.2 

41.3 

0.0 

41.3 

9. 712614 






-3.6 


0.1 




Ax—As 








3. 707429 


Ax—As 








3.694020 


Ax—A3 








4.055122 

-11+12 

Ax 

91 

32 

03.8 

-1.5 

02.3 

0.1 

02.2 

0.000156 

-4+5 

Ax 

26 

40 

23.5 

+0.2 

23.7 

0.0 

23.7 

9.652151 

-7+9 

As 

61 

47 

35.0 

-0.9 

34.1 

0.0 

34.1 

9.945096 






-2.2 


0.1 




Ax-As 








3. 707429 


Ax-Ax 








4.000374 


* For the method of solution of triangles see United States Coast and Geodetic Survey Special Publi¬ 
cation No. 8, p. 6. 






















24 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


STATION A 3 


Position computation, 


a 

Secondl 
angle / 
a 
Aa 


a' 

<t> 

J </> 

<t>' 

§(<f>+<f>') 


1st term 
2d and 3d\ 
terms J 

— Jcf> 


A 2 to A i 
A\ and A 3 
A 2 to A 3 


A 3 to A 2 


60 


+ 


60 


56 


56 


01.089 
56. 720 


60 56 29 


-57.0380 
+ 0.3184 


-56. 7196 
s 

sin a 
A' 

sec <£' 


57.809 
s 

cos a 
B 


AX 


3.932100 
9.990544 
8. 508600 
0. 313737 


2. 744981 

n 

-555.8800 


First angle of triangle 
A 3 


X 

AX 

X’ 


0 

/ 

// 

156 

20 

26.6 

+ 101 

44 

46.6 

258 

05 

13.2 

+ 

8 

05.9 

180 

00 

00.0 

78 

13 

19.1 

30 

43 

52.8 

149 

34 

19. 237 

— 

9 

15. 880 


3.932100 
9.314765 
8. 509299 


1. 756164 


S 2 

sin 2 a 
C 


7.86420 
9. 98109 
1.65750 


9. 50279 

0.3183 
0 . 0001 


149 


7* 2 

D 


AX 

sin i(<f>+<f>') 


—AOC 


2.744981 
9. 941572 


2. 686553 


-485. 89 


25 


3.5122 
2.3224 


5.8346 


03.357 


STATION Ai 


<X 

Secondl 
angle j 
a 
Aa 


a' 



0 

/ 

// 


60 

56 

01.089 

At}> 

— 


55.340 

4’ 

60 

55 

05. 749 




1st term 
2d and 3dl 
terms / 
—A<t> 


A% to A 3 
A 3 and A\ 
A 3 to A 4 


A 4 to A 2 


60 55 33 

n 

+55.2425 
+ 0.0979 


+55.3404 


cos a 
B 


s 

sin a 
A' 
sec 


AX 


3. 694020 
9.972328 
8.508601 
0.313313 

2.488262 

n 

-307. 7953 


First angle of triangle 

A2 X 

AX 

X' 


O 

/ 

H 

258 

05 

13.2 

+ 32 

08 

59.7 

290 

14 

12.9 

+ 

4 

29.0 

180 

00 

00.0 

110 

18 

41.9 

116 

47 

19.1 

149 

34 

19. 237 

— 

5 

07. 795 

149 

29 

11.442 


0.0978 

1 


AX 

sin §(<£+<£') 


—A(X 


2.488262 
9.941507 


2.429769 

// 

-269.01 


3.694020 

$2 

7.38804 



9.538954 

sin 2 a 

9.94466 

h 2 

3.484 

8. 509299 

C 

1.65750 

D 

2.322 

1. 742273 


8.99020 


5.806 


* For an explanation of the forms for computing differences of latitude, longitude, and azimuth see 
United States Coast and Geodetic Survey Special Publication No. 8, pp. 6-11. 

































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


25 


secondary triangulation * 


STATION A 3 


a 

Third! 

angle/ 

a 

ja 

a' 


j(j> 

</>’ 


1 st term 
2 d,3d, and 
4 th 
terms 
— J<f> 


A i to A 2 
A 3 and At 
A i to A 3 

A 3 to A i 


GO 


f»0 


58 

1 


56 


GO 57 57 

n 

+ 118.0821 
+ 0.5258 


+118.6079 
s 

sin a 
A' 

sec tf>' 








O 

t 

n 







336 

18 

08.4 







- 47 

31 

20.7 







288 

46 

47.7 







+ 

10 

24.3 







180 

00 

00.0 







108 

57 

12.0 

tt 








-.1 

56. 416 


A! 



X 

149 

36 

57.360 

58. 608 




JX 

— 

11 

54.003 

, 57.808 
+ 1 


A 3 



X' 

149 

25 

03.357 

s 

4.055122 

S 2 

8.11024 



-h 

2.072 

cos a 

9. 507767 

sin 2 a 

9.95248 

h* 

4.1144 

s 2 sin 2 a 

8.063 

B 

8 . 509295 

C 

1.65837 

D 

2.3221 

E 

6 . 640 

h 

2.072184 


9. 72109 


6.4665 


6 . 775 




0.5261 

0.0003 








-0.0006 






JX 


4.055122 
9. 976241 
8.508600 
0.313737 

JX 

sin £(<£+<£') 

2.853700 


n 


-714.0029 

—jot 


2.853700 
9.941676 


2. 795376 
// 

-624.27 


STATION A< 


a 

Third! 

angle/ 

a 

ja 


a' 


A 3 to A 2 
A\ and A 2 
A 3 to A 4 


Act, 




1 st term 
2d and 3d! 
terms J 
— J<t> 


A 4 to A 3 

O 

60 


60 


57. 809 
52.060 


55 05.749 


A 3 

A< 


X 

JX 
X « 


0 

/ 

// 

78 

13 

19.1 

- 31 

03 

41.3 

47 

09 

37.8 

— 

3 

36.8 

180 

00 

00.0 

227 

06 

01.0 

149 

25 

03.357 

+ 

4 

08.086 

149 

29 

11.443 


O ft 


s 

3.707429 

S 2 

7.41486 



60 56 02 

cos a 

9.832475 

sin 2 a 

9. 73052 

h 2 

4.098 

// 


B 

8 . 509298 

C 

1. 65808 

D 

2.322 

+ 111.9959 

h 

2. 049202 


8 . 80346 


6.420 

+ 0.0639 




0.0636 


+ 112.0598 




3 


s 


3.707429 





sin a 


9. 865259 





A' 


8 . 508601 

JX 

2.394602 



sec <t>' 


0.313313 

sin £(<£+<£') 

9.941540 





2.394602 


2.336142 





// 


n 



JX 

+248.0858 

-ja 

+216.84 




- 1 





































































26 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


List of geographic positions , Turnagain Arm, Alaska. Valdez datum 


Station 

Latitude and 
longitude 

Seconds 
in meters 

Azimuth 

Back azimuth 

To sta¬ 
tion 

Distance 

Logarithm 

A 3 

o t if 

60 56 57.809 

1789.4 

78 13 19.1 

O t ft 

258 05 13.2 

As 

Meters 
8552.6 

3.932100 


149 25 03.357 

50.5 

108 57 11.9 

288 46 47.7 

A\ 

11353.3 

4.055122 ' 

a 4 

60 55 05.749 

177.9 

110 18 41.9 

290 14 12.9 

As 

4943.3 

3.694020 


149 29 11.442 

172.4 

135 33 58.7 

315 27 11.4 

Ai 

10008.6 

4.000374 




227 06 01.0 

47 09 37.8 

As 

5098.3 

3.707429 


DEVELOPMENT OF CONDITION EQUATIONS FOR LATITUDE AND 
LONGITUDE CLOSURES 

After the conditions arising from the closure of triangles and from 
the equality of sides or lengths computed by different routes have 
been satisfied, cases frequently arise where azimuth, latitude, and 
longitude conditions must be satisfied. There is given now a devel¬ 
opment of a form of condition equations that will bring about a 
closure in geographic position. 

Discrepancies in latitude and longitude arise whenever a chain of 
triangulation or a traverse closes on itself. The discrepancies may 
be distributed throughout the whole loop or in a selected portion of 
it, depending upon the circumstances. Of course the most rigid 
adjustment would require the discrepancies to be distributed through¬ 
out the whole chain. At times, however, this would require more 
labor than the importance of the work would justify. Also some 
parts of the loop may be much better determined than other parts, 
in which case the more poorly determined part should be required to 
make up the discrepancies. 

The discussion of the form of equations to be employed to effect 
the closure without discrepancies will be based upon the position 
computation formulae employed by the United States Coast and 
Geodetic Survey i (See United States Coast and Geodetic Survey 
Special Publication No. 8, p. 8.) 

The amount to be distributed being, of course, small compared 
with the total change in latitude and longitude, the only term of the 
latitude computation formula that need be considered is the first 
one. No appreciable changes due to the adjustment will take place 
in the other terms. 

The formation of the equations must always start from a line fixed 
in length and azimuth. If a scheme of triangulation should start 
from a fixed line and run to two points which are fixed in position 
but are not the ends of a single line, then the formation of the equa¬ 
tions for each of the two points must start from the fixed line. 

There are, of course, two elements that enter into the determination 
of the position of any point as computed from a known point; these 












APPLICATION OF LEAST SQUARES TO TRIANGULATION. 27 


are the distance from the known point and the azimuth of the line 
from the known to the unknown point. 

In the triangle 12 3, let 1 and 2 he fixed in position, and let us 
consider what change in the position of 3 will be produced by small 
changes in angles A, B, and C. The length to be carried forward is 

1 to 3. Starting with the length 1 to 2, we have log 1 to 3 =log 1 to 

2 — log sin B + log sin A. The change in length, then, depends upon 
the changes in angles A and B and the change in azimuth of the line 
1 to 3 depends upon the change in angle C. The angles A and B, 
therefore, are called the length angles and angle C the azimuth angle. 

If we can derive a linear expression for the effect of each of these 
separately, the total effect will be the sum of the two. 

Let d x and d B represent the change of the log sin for a change of 
one second in the angles A and B; v x and v B the number of seconds 
change in angles A and B, respectively. Then the change in log sin 
A will equal d x (v A ), and the change 
in log sin B will equal d B (v B ); there¬ 
fore, the change in log 1 to 3 is 
+ (Ll) ~ (^ B ) • The change in the 

logarithm of thefirst term of thelati- 
tude due to the change in length is 
equal to This is 

the change in the logarithm, but for 
convenience of computation it is 
better to determine what change in 
the antilogarithm will be produced 
by this change; or, in other w^ords, ^ 
to determine what this logarithmic 
change will amount to in seconds of arc. From the nature of loga¬ 
rithms, if we know the number to which a given logarithm corre¬ 
sponds, the change in the number due to any small change in the 
logarithm can be found by multiplying the logarithm change by the 
number and dividing by M (the modulus of the common system of 
logarithms). This can also bo shown by differentiation. 

Let y = \og 10 x 

dy= M — 

J X 

Therefore dx = ^ dy, dy being the small change in the log and dx 

the corresponding small change in the number. 

+ must then be multiplied by (<£ B -<£ C )> (m which 

<j) B is the computed latitude of 3 and <j> c is the latitude of 1), and the 
product divided by M; this will give the change in seconds in the 
latitude of 3 due to the change in length of 1 to 3. 


> 5 - 





28 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Next must be considered the change in latitude due to the change 
in the azimuth angle C, If s is the length in meters of 1 to 3, the 
length of the small arc through which 3 turns is equal to s(v c ) arc 1" 
(as ds = rdO for a circle about the origin in polar coordinates), v c 

equals the number of seconds 
N change in angle C, and arc 1 " 
A is included to reduce this angle 
to circular measure. 

Let 3 be the original posi¬ 
tion of 3 and 3' the position 
due to a small rotation of 1 
H to 3 about 1. 

n 

3 to 3' =sv c arc 1" 

FlG - 3 - The azimuth of 3 to 3' is 90° 

+ «. The change in latitude due to s(v c ) arc 1" is equal to — s(v c ) 
arc 1" cos (90° + <*) times the B factor in the position computation, 
= + s(v c ) arc 1" sin a B 
= + (v c ) arc 1 " Bs sin a 
But I B —= s sin ot A' sec <£>' 

Therefore s sin a = 



A' sec (£>' 


B arc r 


7 (^B Ic) 0^c)« 


Therefore the change in latitude = jT- SGG( y 

In a similar way the change in longitude due to a change in length is, 

and the change in longitude due to the change in azimuth is, 
s(v c )arc 1" sin (90° + **) A' sec </)'= + ( v c ) arc 1" s cos a A ' sec <£>' 
— s cos a’B = 4 > }i —(j>c (neglecting the small terms) 

8~ 


COS OL = — J 


B 


Therefore change in longitude = — j—^ ° c ^ alc 1 - (<£ B — <£ c ) (v c ). 

The usage is to point off the log change for one second of arc as an 
integer in the sixth place of logarithms; therefore as a number the 
d d 

tabular difference = and • 

The total change in seconds of latitude in the triangle is 


10 # M I 


+ - 3 b (v, 




. B arc 1" w N * 

± A'sec<j>' Wb 


The total change in longitude is, 


10 6 M [ 


+ d A (v A )-d B (v B ) T 


A' sec cj)' arc 1' 


(^B <£ C ) (V c ) • 


The upper sign beine used for a right azimuth angle, the lower sign for a left. 













APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


29 


In this way the change could be determined for each triangle in 
the chain and the sum placed equal to the discrepancy, but this 
would require a very great amount of work. 

If any change takes place in the first triangle while the remaining 
triangles are for the moment supposed to remain fixed, this length 
change and azimuth change will affect not only this triangle, but 
will persist in each succeeding triangle. As a consequence the 
change of length and azimuth in the first triangle will be felt in the 
computation of every point after it in the chain. Let cj> n and X n be 
the computed (j> and X of the end point. The change in the first 
triangle will apply not merely to — etc., but to — etc. 

Therefore the change in the final position due to the changes in the 
first triangle is, for latitude, 


10 6 M L 


+ ^A.( V a) ~ 



B c ar c 1 " 
± 37sec f n 


(4-A c ) M* 


and for longitude, 


10 6 M 




A n sec (j) n arc 1 " 


B c 


(<£ n -<£c) (v c )* 


In the same way the change in the final position due to changes 
in the second triangle can be determined, and so on through the 
whole chain. Each triangle will have an A, B , and C angle, A 
being the length angle next to the known side; B, the one opposite 
the known side; and C, the azimuth angle. 

The equations will finally stand 


0 = + (dn — + A tl ^o 6 M 

0 = + (X n — Xn') + 2 j”jQ 6 dM — pQ6“j|- *W] 
A n sec <j) n arc l f 




B c 


(4>n-4> c)(V c )* 


<J) n is the computed latitude of the final point and (j) n f , the fixed lati¬ 
tude; so also for X n and X n '. 

It is exact enough to take (f) n - (f> c and X n -X c to minutes and tenths 
of a minute, so that it is advisable to divide the equations by 60 since, 
as they stand, <j> n ~ etc., are in seconds. Also it is best to mul¬ 
tiply through by 10 6 M to remove this factor from the denominator 
of the first summation. 


# Upper sign for right azimuth angle, lower for left. 














30 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Then we have 


0=+gQ 10>(4> n -4>n')"+2[(<l> n —0c) 7 {(p n <f> c ) 

+ 2-± 10 6 ^Jsec'k (4_Ac) ' (Vc) ' * 

0= 10 6 (.K.-Xn')" + mn-KY KM-iK-W o\K)] 

+i , T 10 6 M An scc ^ arc *" (4>»-4>oYM. * 


The A and B factors change so slowly that for any chain they can 
be taken for the mean <f> and also sec (p n can be used in the same way. 
A table can then be prepared for functions designated as oq and a 2 
and defined as follows: 


, 1 , r B arc 1" 

a, = +10 6 M r-j -x 

. 1 A sec (p 


a 9 =-10 6 M 


A sec (j> arc 1' 

~~B~ 


the A and B factors and the (f> being used at a convenient interval. 
A table has been computed for latitudes starting at 24° for intervals 
of 4° up to 56°. The minus sign is used with a 2 in order that the 
same sign can be used on the directions of the azimuth angle for 
both latitude and longitude equations. If the discrepancy to be 
made up by the adjustment is large, or if the chain extends over a 
great distance of latitude, it would be best to compute the values of 
a t and a 2 using A n and cp n and the B for the mean <f>. 

If the chain to be adjusted extends principally east and west, in 
place of — c a summation of the first terms (Ji) in the position 

computations should be used. I c Ji would then replace (f> n — <p c , the 

sign being used that would conform to (j> n —<p c . These quantities 
should then be used throughout in forming the equations. 

If the latitude and longitude equations are to be included in the 
main adjustment and the equations all solved simultaneously, the 
computation of the positions through the chain must be made with 
one length carried through the figures by means of the observed 
plane angles; that is, the angles as observed each diminished by J of 
the spherical excess of the triangle. This could be done by carrying 
the length through a selected chain of triangles and then computing 
each of the various positions over a single line. Both lines of the 
triangle could not be used because the observed plane angles must 
be used in carrying the length and, under ordinary circumstances, 
the triangle would not be closed. To obviate this difficulty, it is 
best to use only the observed A and B angles and to conclude the C 
angle, using, of course, the concluded correction symbols on this 


* Upper sign for right azimuth angle, lower for left. 







APPLICATION OF LEAST SQUARES TO TRIANGULATION. 31 

angle. This method gives a much more reliable determination of 
the discrepancy, as it furnishes a check on each position, and thus 
prevents a mistake being left in the computation. If the figure 
adjustment is carried out first, there is no need to follow this method 
as the triangles would then be closed. In this case it is the general 
custom of the United States Coast and Geodetic Survey to choose 
the best chain of triangles and to form the equations through them, 
using the angle method in place of the direction method. Equations 
with absolute terms equal to zero must be included for the various 
triangles in order to hold them closed; also, if a length.equation is 
included in the figure adjustment, it must be retained with zero 
discrepancy to hold the length. If the figure ends on a fixed line 
and a length equation is not put in the figure adjustment, the dis¬ 
crepancy must be put on the length equation used with the latitude 
and longitude equations. After adjustment is made for these final 
discrepancies the cross lines are computed by two sides and the 
included angle. 

The best results are probably obtained by the solution of all the 
equations at once, but this entails so much work that the angle 
method is often used in chains of minor importance. 

We have finally: 

M 

j= 10 6 = 7238.24 
oU 

0= +7238.24(0,— 0»O" + -?[(0»- 

— (0w~ 0c) ' ^b( v b)] + ^ ± a iWn — A c )'(v c ). 

0 =+ 7238.24 (4 - W" + A [(4 - 4)'<*a(v a ) 

— — + 2 0c) r ( V c)* 

In the equations v A , v B , and v c would be replaced by their correc¬ 
tion symbols, care being taken to use v c = -v x — v B7 if the azimuth 
angle has been concluded in carrying the position computation 
through the chain. 

If an azimuth equation occurs, the constant term must be corrected 
by +(4-; n /)xsine of the mean 0, this being the amount that the 
azimuth will change from the changes in the back azimuths due to 
the changes in longitude. 

It should be noted that whenever a discrepancy of position is 
adjusted into a section of a loop, an external condition is placed upon 
the chain, as at best only part of this discrepancy is due to errors in 
the chain, the rest being due to the remainder of the loop. It is 
necessary to hold some parts of the triangulation fixed; otherwise 
when a loop closure is put in it would frequently be necessary to 
readjust nearly all of the triangulation of the country. The result 
is, however, that some chains of triangulation, excellent in them¬ 
selves, get some rather large corrections due to the position closure. 

91865°—15-3 


32 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

EQUATIONS IN A NET 

In the adjustment of a quadrilateral, use is made of the two kinds 
of condition equations that are necessary for the adjustment of any 
figure that does not contain external conditions such as length, azi¬ 
muth, or loop closure. In fact most figures can be broken up into 
successive quadrilaterals. In forming the length equation, use is 
made of the two length angles in the various triangles passed through. 
In fig. 5, on page 37, the length angles are lettered A and B. The 
angle omitted is the azimuth angle of the given triangle. The log 
sin of the A angle is added to the first length and the log sin of the 
B angle to the final length. So with all of the triangles through 
which the length is carried. The discrepancy is found and the equa¬ 
tion formed in the same way as in the case of an ordinary side equa¬ 
tion. See the formation of the length equation on page 37. If the 
spherical angles are used a correction for arc to sine must be applied 
to each length. (See the table of these corrections in Special Pub¬ 
lication No. 8, p. 17.) 

An azimuth equation is formed by adding algebraically to the 
first azimuth the various azimuth angles up to the second line fixed 
in azimulfti. When passing from one end of a line to another, the 
.azimuth difference due to convergence of the meridians, must be 
applied as determined in the computation of positions. The alge¬ 
braic sum of the v’s upon these angles must make up the discrepancy 
between the computed and fixed azimuths. See the computation 
on page 38. 

The determination of the exact number of side and angle equa¬ 
tions in a net and the manner in which they come in, is one of the 
difficulties encountered by a beginner in the adjustment of trian¬ 
gulation. This is especially true if the net is somewhat complicated. 
The best method for this determination is to plot the figure point by 
point. By plotting the triangle Tower, Turn, and Dundas, in the 
figure on page 34 one angle equation is determined. Add Lazaro 
by the lines Lazaro to Turn and Lazaro to Tower. This gives another 
angle equation, making two. Another angle equation and a side 
equation are obtained by putting in the line Lazaro to Dundas. 
This makes a total of three angle equations and one side equation 
for the quadrilateral, just as it should be. Next plot Nichols by the 
lines Nichols to Lazaro and Nichols to Tower; this is a closed triangle 
and gives a fourth angle equation. Put in Tow Hill by the lines 
Tow Hill to Nichols and Tow Hill to Lazaro; this does not give an 
angle equation as it is not a closed triangle. Draw the line Tow 
Hill to Tower; this gives a second side equation. In this way one 
can continue through the whole figure. If a full line Nichols to 
Turn were in the figure, it would give another angle and another side 


APPLICATION OF LEAST SQUARES TO TRIANGULATION. 33 


equation. The angle equation added would have to include the 
directions on this line as would also the side equation. This method 
shows at once where the equations come in and what new v’s must 
appear in the equations. 

Lines sighted over in only one direction have no effect on the 
number of angle equations. If the closed part of the figure is plotted, 
omitting all of the extra lines—that is, putting in each station with 
only two lines from those already plotted, a closed framework of the 
figure will be formed. The first triangle requires three lines, those 
after the first require two lines. The number of angle equations in 
the framework of the figure is thus equal to the number of lines in 
the figure minus the number of stations plus one. Every full line 
added to this framework gives another angle equation. Therefore, 
the whole number of angle equations in a net is equal to the whole 
number of full lines minus the number of occupied stations plus one. 

The lines sighted over in one direction have the same effect on the 
number of side equations that the full lines have. If the full frame¬ 
work of the figure is plotted with two lines to each station from those 
already determined, no side equation will as yet appear in the figure. 
Every extra line put in gives a side equation. The first triangle 
fixes three stations; the stations after these require two lines to be 
used in plotting them. Thus the number of lines needed to plot the 
framework is equal to twice the number of stations minus three. 
The full number of side equations will then be equal to the number 
of all the lines minus twice the number of all the stations plus three. 

Let n = total number of lines. 

ti' = number of lines sighted over in both directions. 

S = total number of stations. 

S' = number of occupied stations. 

Then 

The number of angle equations in a net = 7i' — S' +1. 

The number of side equations in a net = 77 —2 $ + 3. 

These formulas should be used to check the number determined by 
directly plotting the figure. 

In figure 4 on page 34, 

n = 41 
n' = 38 
S =18 
S' = 17 

Therefore number of angle equations = 38 — 17 +1 = 22. 

number of side equations = 41—36 + 3= 8. 

For convenience in solution it is best to use triangles with the 
larger angles for the angle equations, reserving the small angles to 
be used in the side equations. This will keep the large coefficients 


34 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

in the side equations from appearing on the same directions as are 
used in the angle equations and will aid in the solution of the normals. 
The small angles need to appear in the side equations, as their tabular 
differences are proportionally much less affected by the dropping of 
decimal places than are those of the larger angles. 

ADJUSTMENT OF A FIGURE WITH LATITUDE, LONGITUDE, AZIMUTH, 
AND LENGTH CLOSURE CONDITIONS 



'dr' 

Ton'S//// 

Fig. 4 . 

In this figure, in addition to the angle, side, and length conditions, 
there are included conditions for azimuth, latitude, and longitude. 




APPLICATION OP LEAST SQUARES TO TRIANGULATION. 


35 


Angle equations 

0=- 5.5- (1)+ (2)- (4)+ (6) — (lO)-f-(ll) 

0=+ 4.0- (1)+ (3)- (5)+ (6) t (12)+(13) 

0=+12.0- (2)+ (3)- (9)+(10)-(12)+(14) 

0=- 8.0- (8)+ (9)-(14)+(16)-(25)+(26) 

0=- 2.9-(16)+(17)-(23)+(25)-(31)+(32) 

0=— 0.0—(16)+(18) — (24)+(25)—(33)+(34) 

0= - 3.2-(23)+(24) -(30)+(32) -(34)+(36) 

0=— 2.3—(17)+(19) —(29)+(31)—(36)+(37) 

0=— 3.2 —(17)+(21) —(28)+(31) —(42)+(45) 

0= - 3.2 -(19)+(21)+(36) -(38) -(42)+(44) 

0=+ 5.0 —(21)+(22) —(40)+(42)—(49)+(52) 

0=— 1.0 —(20)+(22)—(47)+(48)—(49)+(51) 

0=+ 3.7 — (40)+(43) — (46)+(47)—(51)+(52) 

0=+ 5.2 —(40)+(41) —(50)+(52)—(56)+(58) 

0=— 2.5—(39)+(40)—(52)+(55)—(59)+(61) 

0= + 2.8 —(50)+(55) —(57)+(58)—(59)+(60) 

0=— 5.1—(53)+(55)—(59)+(62) —(67)+(68) 

0=— 1.7—(54)+(55)—(59)+(63)—(71)+(72) 

0=+ 2.1 —(62)+(63)—(66)+(67)—(71)+(73) 

0=+ 2.3 —(63)+(65) —(70)+(71)—(74)+(75) 

0=+ 6.1 —(64)+(65)—(74)+(76)—(77)+(78) 

0=+ 0.1 —(69)+(70) —(75)+(76)—(77)+(79) 

Azimuth equation 

0=(—7.1*—0.2)—(l)+(2)+(9)—(10)—(14)+(21)+(40) —(42) 

-(52)+(55)-(59)+(63)+(70)-(71)-(75)+(76) 

Computation of correction to azimuth constant: 

log 0.277=9. 442 
log sin. mean $=9. 912 
log correction=9. 354 
correction = —0. 2 


Side equations 


Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

- 9+11 
-12+13 
-1+2 

93 11 39.1 

25 52 38.1 

24 17 25.4 

9. 9993248 

9.6399291 

9.6142236 

-0.12 
+4.34 
+4.67 

-13+14 

-1+3 

-10+11 

16 37 43.4 
110 36 08.7 

42 00 30.0 

9.4566222 • 
9.9712965 

9.8255810 

+7.05 
-0.79 
+2.34 

9.2534775 

9.2534997 


0= —22.2—5.46(l)+4.67(2)+0.79(3)+0.12(9)+2.34(10) —2.46(11)—4.34(12)+11.39(13) 
-7.05(14) 


- 7+ 9 

111 

09 

20.5 

9.9696969 

-0.81 

/ + 7- 9 
\ +14-15 

}m 

40 

38.8 

9.5674745 

+5.30 

+15-16 

+25-27 

} 36 

08 

04.3 

9.7706188 

+2.88 

-25+27 

89 

23 

18.6 

9.9999753 

+0.02 

-25+26 

30 

04 

51.8 

9.7000325 

+3.64 

-8+9 

48 

16 

10.2 

9.8729041 

+1.88 





9.4403482 






9.4403539 



0=—5 . 7 — 4 . 49(7)+1.88(8)+2.61(9)—5.30(14)+8.18(15)—2.88(16)—0.74(25)+3.64(26) 
-2.90(27) 


* See computation on p. 38. 


























3G COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Side equations —Continued 


Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

Symbol 

Angle 

Logarithm 

Tabular 

differ¬ 

ence 

-16+18 

-23+24 

-30+31 

38 29 18.8 

27 51 39.2 

9 31 16.8 

9. 7940405 

9. 6696203 

9.2185744 

+ 2.65 
+ 3.98 
+12.55 

-24+25 

-30+32 

-17+18 

12 35 33.3 
126 31 06.3 

15 56 21.5 

9.3384902 

9.9050754 

9.4387305 

+9.43 
-1.56 
+7.37 

8.6822352 

8.6822961 


0= —60.9—2.65(16)+7.37(17)-4.72(18) -3.98(23)4-13.41(24) -9.43(25) -14.11(30) 
4-12.55(31)4-1.56(32) 


-17+19 

35 17 43.3 

9.7617712 

+ 2.98 

-29+31 

16 21 23.9 

9.4496564 

+7.17 

-28+29 

9 58 42.9 

9.2387486 

+11.97 

-44+45 

23 37 23.0 

9.6028386 

+4.81 

-42+44 

51 05 11.2 

9.8910323 

+ 1.70 

-19+21 

43 39 34.4 

9.8390831 

+2.21 



8.8915521 




8.8915781 



0=-26.0-2.98(17)4-5.19(19)-2.21(21)-11.97(28)4-19.14(29)-7.17(31)-1.70(42) 
4-6.51(44)-4.81(45) 


-49+52 

89 47 52.8 

9.9999973 

+0.01 

-21+22 

26 22 55.0 

9. 6477280 

+4.25 

-20+21 

20 37 18.2 

9. 6513730 

+4.20 

-46+48 

105 41 48.0 

9.9834943 

-0. 59 

-46+47 

33 20 40.5 

9.7401044 

+3.20 

-51+52 

35 09 14.0 

9.7602524 

+2.99 



9.3914747 




9.3914747 



0=4-0.0-4.20(20)4-8.45(21)-4.25(22)-3.79(46)+3.20(47)4-0.59(48)-0.01(49) 
4-2.99(51)-2.98(52) 


-39+41 

105 37 20.7 

9.9836521 

-0.59 

-60+61 

22 08 21.3 

9.5761788 

+5.17 

-59+60 

27 49 50.7 

9.6691879 

+3.99 

-50+55 

125 14 18.4 

9.9120934 

-1.49 

-50+52 

37 39 24.5 

9.7859918 

+2.73 

-40+41 

63 10 28.9 

9.9505530 

+1.07 



9.4388318 




9.4388252 



0=+6.6+0.59(39)4-1-07(40)-1.66(41)-4.22(50)-f-2.73(52)4-1.49(55)-3.99(59) 
+9.16(60)-5.17(61) 


-71+73 

59 25 24.7 

9.9349784 

+1.24 

-62+63 

43 56 28.3 

9.8413093 

+2.19 

-59+62 

58 43 17.2 

9.9317900 

+1.28 

-53+55 

59 03 08.4 

9.9333037 

+ 1.26 

-53+54 

18 17 51.0 

• 

9.4968619 

+6.37 

-72+73 

22 50 29.2 

9.5890357 

+5.00 



9.3636303 




9.3636487 



0= —18.4—5.11(53)+6.37(54) —1.26(55) —1.28(59)+3.47(62)—2.19(63) —1.24(71) 
+5.00(72)-3.76(73) 


-74+76 

94 10 29.2 

9. 9988462 

-0.15 

-64+65 

50 11 06.3 

9.8854273 

+1.76 

-63+64 

28 13 32.1 

9.6748099 

+3.92 

-69+71 

102 20 31.6 

9.9898451 

-0. 46 

-69+70 

61 25 13.8 

9.9435708 

+1.15 

-75+76 

33 30 23.0 

9. 7419627 

+3.18 



9.6172269 




9.6172351 



0= —8.2—3.92(63)+5.68(64) —1.76(65) —1.61(69)+1.15(70)+0.46(71)+0.15(74) 
+3.18(75)—3.33(76) 






































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


37 


Length equation 


Symbol 





Tabular 






Tabular 

- 

Angle 

Logarithm 

differ- 

Symbol 


Angle 

Logarithm 

differ- 






ence 






ence 





-6 






-1 


Turn-Dundas 


4.266771 


Ham-South Twin 

3.898371 


-4+6 

113 

41 

59.6 

9.9617359 

-0.92 

-10+11 

40 

00 

30.0 

9.8255810 

+2.34 

-2+3 

86 

18 

43.3 

9.9990997 

+0.14 

-12+14 

42 

30 

21.5 

9.8297327 

+2.30 

-7+9 

111 

09 

20.5 

9.9696969 

-0.81 

/ + 7- 9 
\ +14-15 

} 21 

40 

38.8 

9.5674745 

+5.30 

+15-16 

+25-27 

} 36 

08 

04.3 

9.7706188 

+2.88 

-25+27 

89 

23 

18.6 

9.9999753 

+0.02 

-23+25 

40 

27 

12.5 

9.8121311 

+2.47 

-31+32 

116 

59 

49.5 

9.9498922 

-1.07 

-28+31 

26 

20 

06.8 

9.6470132 

+4.25 

-42+45 

74 

42 

34.2 

9.9843478 

+0.58 

-21+22 

26 

22 

55.0 

9.6477280 

+4.25 

-49+52 

89 

47 

52.8 

9.9999973 

+0.01 

-39+40 

42 

26 

51.8 

9.8292505 

+2.30 

-59+61 

49 

58 

12.0 

9.8840631 

+ 1.77 

-54+55 

40 

45 

17.4 

9.8147959 

+2. 44 

-71+72 

36 

34 

55.5 

9.7752272 

+2.84 

-63+65 

78 

24 

38.4 

9.9910544 

+0. 43 

-74+75 

60 

40 

06.2 

9.9404164 

+ 1.18 

-69+70 

61 

25 

13.8 

9.9435708 

+ 1.15 

-77+79 

85 

04 

23.4 

9.9983924 

+0.18 





2.6534656 






2.6534708 



0=—5.2—0.14(2)+0.14(3)+0.92(4)—0.92(6)—4.49(7)+4.49(9)+2.34(10)—2.34(11) 
+2.30 (12)—7.60 (14)+8.18 (15)-2.88(16)-4.25(21)+4.25(22)-2.47(23)+5.37(25) 
-2.90 (27)-4.25 (28)+3.18 (31)+1.07(32)-2.30(39)+2.30(40)+0.58(42)-0.58(45) 
+0.01 (49)-0.01 (52)-2.44 (54)+2.44(55)+1.77(59)-1.77(61)-0.43(63)+0.43(65) 
-1.15 (69)+1.15 (70)+2.84 (71)-2.84(72)+1.18(74)-1.18(75)+0.18(77)-0.18(79) 



Fig. 5. 

























38 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Formation of azimuth equation 



O 

/ 

// 


o 

/ 

// 

Turn-Dundas. 

357 

33 

07.7 

Cat-Round. 

.. 109 

42 

29.5 

-1+2. 

24 

17 

25.4 

-52+55.. 

87 

34 

53.9 

Turn-Tower. 

21 

50 

33.1 

Cat-Beaver. 

.. 197 

17 

23.4 

Azimuth difference*.... 

-180 

07 

07.6 

Azimuth difference... 

.. +180 

01 

40.2 

Tower-Turn. 

201 

43 

25.5 

Beaver-Cat. 

17 

19 

03.6 

+9-10. 

51 

11 

09.1 

-59+63. 

.. 102 

39 

45. 5 

Tower-Lazaro. 

150 

32 

16. 4 

Beaver-Liin. 

.. 119 

58 

49.1 

Azimuth difference. 

-180 

14 

01. 2 

Azimuth difference... 

.. -180 

05 

20.3 

Lazaro-Tower. 

330 

18 

15.2 

Lim-Beaver. 

.. 299 

53 

28.8 


r ioi 

38 

55.1 

+70-71. 

40 

55 

17.8 

-14+21. 

22 

32 

57.3 

Lim-South Twin. 

.. 258 

58 

11.0 


1 78 

57 

17. 7 

Azimuth difference... 

.. +180 

06 

48.1 

Lazaro-Round. 

173 

27 

25.3 

South Twin-Lim. 

79 

04 

59.1 

Azimuth difference. 

-180 

01 

37.8 

-75+76. 

33 

30 

23.0 

Round-Lazaro. 

353 

25 

47.5 

South Twin-IIam- 

. 112 

35 

22.1 

+40-42. 

63 

49 

17. 6 

Fixed azimuth. 

.. 112 

35 

29.2 

Round-Cat. 

289 

36 

29.9 





Azimuth difference. 

+180 

05 

59. 6 

Discrepancy... 



-7.1 

Cat-Round. 

109 

42 

29. 5 






Preliminary computation of triangles 


Desig¬ 

nation 

of 

angle 

Symbol 

Station 

Observed 

angle 

Cor¬ 

rection 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane angle 

Logarithm 




or rr 



rr 

or rr 




Tum-Dundas 






4 . 266771 

B 

-10+11 

Tower 

42 00 30.0 



0.2 

29.8 

0.1744195 

C 


Turn 

30.9 



0.2 

24 17 30.7 

9. 6142483 

A 

-4+6 

Dundas 

113 41 59.6 



0.1 

59.5 

9. 9617359 







0.5 





Tower-Dundas 


• 



4.0554388 



Tower-Turn 






4.4029264 



Turn-Tower 






4.4029264 

B 

-12+14 

Lazaro 

42 30 21.5 



0.6 

20.9 

0.1702686 

A 

-2+3 

Turn 

86 18 43.3 



0.7 

42.6 

9. 9990996 

C 


Tower 

57.1 



0.6 

51 10 56.5 

9. 8916183 







1.9 





Lazaro-Tower 






4.5722946 



Lazaro-Tum 






4.4648133 



Lazaro-Tower 






4.5722946 

B 


Tow Hill 

38.8 



2.2 

21 40 36.6 

0.4325371 

C 

-14+15 

Lazaro 

47 10 07.2 



2.2 

05.0 

9.8653119 

A 

-7+9 

Tower 

111 09 20.5 



2.1 

18.4 

9. 9696986 







6.5 





Tow Hill-Tower 





4.8701436 



Tow Hill-Lazaro 





4. 9745303 



Lazaro-Tow Hill 





4. 9745303 

B 

-25+27 

Nichols 

89 23 18.6 



3.6 

15.0 

0.0000248 

C 

-15+16 

Lazaro 

54 28 47.9 



3.6 

44.3 

9. 9105723 

A 


Tow Hill 

04.3 



3.6 

36 08 00.7 

9. 7706083 







10.8 





Nichols-Tow I 

ill 





4. 8851274 



Nichols-Lazaro 

i 




4. 7451634 


* See position computation, p. 40. 
























































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


39 


Desig¬ 

nation 

of 

angle 


B 

C 

A 


B 

C 

A 


B 

A 

C 


B 

C 

A 


B 

C 

A 


B 

A 

C 


Preliminary computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Cor¬ 

rection 

Spher¬ 

ical 

angle 

Sphei- 

lcal 

excess 

Plane angle 

Logarithm 



O / // 



ft 

o / ft 



Lazaro-Nichols 






4. 7451634 

-31+32 

Ken 

116 59 49.5 



0.8 

48.7 

0.0501070 


Lazaro 

60.2 



0.7 

22 32 59.5 

9. 5.837.509 

-23+25 

Nichols 

40 27 12.5 



0.7 

11.8 

9. 8121294 






2.2 




Ken-Nichols 






4.3790213 


Ken-Lazaro 






4. 6073998 


Lazaro-Ken 






4.6073998 

-42+45 

Round 

74 42 34.2 



0.6 

33.6 

0.0156525 


Lazaro 

20.9 



0.7 

78 57 20.2 

9.9918811 

-28+31 

Ken 

26 20 06.8 



0.6 

06.2 

9.6470108 






1.9 




Round-Ken 






4. 6149334 


Round-Lazaro 






4.2700631 


Lazaro-Round 






4.2700631 

-49+52 

Cat 

89 47 52.8 



0.2 

52.6 

0.0000027 

-21+22 

Lazaro 

26 22 55.0 



0.1 

54.9 

9.6477276 


Round 

12.6 



0.1 

63 49 12.5 

9. 9529926 






0.4 




Cat-Round 






3. 9177934 


Cat-Lazaro 






4.2230584 


Cat-Round 






3. 9177934 

-59+61 

Beaver 

49 58 12.0 



0.1 

11.9 

0.1159371 


Cat 

56.4 



0.0 

87 34 56.4 

9. 9996132 

-39+40 

Round 

42 26 51.8 



0.1 

51.7 

9. 8292503 






0.2 




Beaver-Round 





4.0333437 


Beaver-Cat 






3.8629808 


Beaver-Cat 






3. 8629808 

-71+72 

Lim 

36 34 55.5 



0.0 

55.5 

0.2247728 


Beaver 

47.2 



0.1 

102 39 47.1 

9. 9893057 

—54+55 

Cat 

40 45 17.4 



0.0 

17.4 

9. 8147959 






0.1 




Lim-Cat 






4.0770593 


Lim-Beaver 






3. 9025495 


Beaver-Lim 






3. 9025495 

-74+75 

South Twin 

60 40 06.2 



0.0 

06.2 

0.0595836 

-63+65 

Beaver 

78 24 38.4 



‘ 0.1 

38.3 

9. 9910543 


Lim 

15.5 



0.0 

40 55 15.5 

9. 8162528 






0.1 




South Twin-L 

lm 





3. 9531874 


South Twin-Beaver 

1 





3. 7783859 




































40 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation , 


STATION TOWER 


Second angle 


a 

dot 


(f> 

dtp 

<P' 

S 

COS a 
B 


1st term 
2d term 


3d and 4th 1 
terms / 

—d<j> 

§(<P+<P') 


Turn to Dundas 
Dundas and Tower 

Turn to Tower 


Tower to Turn 


54 


48 

12 


54 | 

4. 4029264 
9. 9676416 
8. 5097251 


35 


06.742 
39.419 


27.323 


2. 8802931 
// 

+759. 0896 
+ 0.3175 


+ 759. 4071 
+ 0.0115 


+ 759.4186 

O / // 

54 41 47 


s 2 

sin 2 a 
C 


3d term 
4th term 


sin a 
A' 

sec <f>' 


dX 


First angle of triangle 
Turn 


8. 80585 
9.14128 
1. 55459 


Tower 


(W 


9. 50172 


+0.0134 
-0. 0019 


-6 

4. 4029264 
9. 5706383 
8. 5087480 
0. 2370138 


2. 7193259 

tt 

+523.9935 


Arg. 

s 

dX 


Corr. 


X 

dX 
X' 

5. 7609 
2. 3672 


8.1281 


-11 

+ 5 


- 6 


0 

357 
+ 24 

t 

33 

17 

ft 

07. 7 

30.9 

21 

50 

7 

38.6 

07.6 

180 

201 

42 

43 

00 

31.0 

30.0 

130 

+ 

56 

8 

04. 052 

43. 993 

131 

-h 
s 2 sin 
E 

2 (x 

04 

2. 

7. 

6. 

48. 045 

8803 

9471 

4574 


7. 2848 


dX 

sin 

sec i(d<f>) 


—da 


2. 7193259 
9.9117440 
7 


2. 6310706 
// 

+427. 63 


STATION LAZARO 


O 

21 

+ 86 

t 

50 

18 

ft 

38.6 

43.3 

108 

09 

21.9 

— 

21 

10.7 

180 



287 

48 

11.2 

42 

30 

21. 5 

130 

56 

04. 052 

+ 

25 

54. 244 

131 

21 

58. 296 
-1 

-h 

2. 4681 


Second angle 


a 

da 


dtp 

<P' 


s 

COS a 

B 


1st term 
2d term 


3d and 4th \ 
terms / 

—dtp 
h(tp+tp') 


Turn to Tower 
Tower and Lazaro 

Turn to Lazaro 


Lazaro to Turn 


54 


+ 


48 

4 


54 


4. 4648133 
9. 4936068 
8. 5097251 


52 


06. 742 
51.101 


57. 843 


First angle of triangle 
Turn 


2. 4681452 

tt 

-293. 8632 
+ 2.7534 


-291.1098 
+ 0.0085 


-291.1013 

O / // 

54 50 32. 3 


s 2 
sia 2 . 
C 


3d term 
4th term 


s 

sin a 
A' 

sec tp' 


dX 


8.92963 
9. 95564 
1. 55459 


Lazaro 


Gg ) 2 


0. 43986 

tt 

+0.0020 

+0.0065 


+27 
4.4648133 
9.9778202 
8. 5087409 
0.2401420 


3.1915191 

it 

+ 1554.2441 


Arg. 

s 

d X 


Corr. 


/ 

dX 

X' 


4. 9281 
2. 3672 


7.2953 


-15 

+42 


+ 27 


1 sin 2 < 
E 


8. 8853 
6. 4574 


7. 8108 


dX 

sin h(<P+<P’) 
sec i(dtp) 


—da 


3.1915191 
9.9125251 


3.1040442 

r r 

+ 1270. 70 


































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


41 


'primary triangulation 


STATION TOWER 


Third angle 


a 

Aa 


<f> 

A<f> 

V 

s 

COS a 

B 

h 


1st term 
2d term 


3d and 4th \ 
terms / 

§(<£ + <£') 


Dundas to Turn 
Tower and Turn 

Dundas to Tower 


Tower to Dundas 

O t 

54 38 

2 


09. 559 
42. 236 


54 

4.0554388 
9.6439767 
8.5097372 


35 


27.323 


2.2091527 

It 

+ 161.8649 
+ 0.3708 


+ 162.2357 

+ 0 . 0001 


+ 162.2358 

O / It 

54 36 48. 4 


s 2 

sin 2 a 
C 


3d term 
4-th term 


s 

sin a 
A' 

sec <j>' 


AX 


8.11087 
9. 90630 
1. 55194 


Dundas 

Tower 


(^) 2 


9. 56911 

It 

+0.0006 
-0.0005 


+3 

4. 0554388 
9. 9531493 
8.5087480 
0. 2370138 


2. 7543502 

n 

+568.0025 


Arg. 

s 

AX 


Corr. 


X 

AX 


4.4203 
2.3681 


6.7884 


-2 
+ 5 


+3 


o 


t 

n 

177 

-113 

33 

41 

43.6 

59.6 

63 

51 

7 

44.0 

43.1 

180 

243 

44 

00.9 
+ .1 

130 

+ 

55 

9 

20. 042 
28.003 

131 

04 

48.045 

-h 

s 2 sin c 
E 

a 

2. 2091 
8.0172 

6. 4528 



6. 6791 

AX 

sin h(<t>+</>') 
sec h(A<t>) 

2. 7543502 

9.9112981 



2. 6656483 




n 

—Aa 

+463.07 


STATION LAZARO 











O 


1 

II 

a 

Tower to Turn 






201 

43 

31.0 

Third angle 

Lazaro and Turn 





- 51 

10 

57.1 

a 

Tower to Lazaro 





150 

32 

33.9 

Aa 









— 

14 

01. 2 










180 




a' 

Lazaro to Tower 





330 

18 

32.7 


O 


f 

If 









54 


35 

27. 323 


Tower 

X 

131 

04 

48.045 

<f> 

+ 


17 

30. 520 



AX 

+ 

17 

10. 250 

d<P 

54 


52 

57. 843 

Lazaro 

A' 

131 

21 

58. 295 









€ 





s 

4.5722946 

$ 2 

9.14459 



-h 


3. 0219 

COS a 

9. 9398800 

sin 2 a 

9. 38353 

(W 

6.0428 

s 2 sin 2 a 

8. 5281 

B 

8. 5097404 


C 

1. 55122 

D 

2. 3683 

E 


6. 4516 

h 

3.0219150 



0.07934 


8.4111 



8.0016 

1st term 

-1051. 7559 

3d term 

+0.0258 







2d term ' 

+ 1.2004 

4th term 

+0.0100 








-1050.5555 











3d and 4 th \ 

+ 0.0358 


s 

— t 

4. 5722946 







terms / 



sin a 

9.6917656 

Arg. 


AX 


3. 0129424 

—A<f> 

-1050. 5197 

J 


8. 5087409 

s 

-25 

sin £(<£+<£') 

9. 9119609 



II 

sec <t>' 

0.2401420 

AX 

+ 18 

sec l(A<fi) 




54 44 12.6 



3.0129424 

Corr. 

- 7 



2.9249033 




AX 

+ 1030.2498 


• 

—Aa 

+841.20 
































































42 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Preliminary position computation , 


STATION TOW HILL 








0 

f 

ft 

a 

Lazaro to Tower 



330 

18 

32.7 

Second angle 

Tower and Tow Hill 



+ 47 

10 

07.2 

a 

Lazaro to Tow Hill 



17 

28 

39. 9 

Act 






— 

21 

07.7 







180 



a 

Tow Hill to Lazaro 



197 

07 

32.2 





First angle of triangle 

21 

40 

38. 8 

<t> 

O 

54 

f 

52 

ft 

57. 843 

Lazaro 

X 

131 

21 

58. 295 

A<f> 

- 

48 

31. 870 


JX 

+ 

25 

57. 539 

<y 

54 

04 

25.973 

1 

Tow Hill 

X' 

131 

47 

55. 834 


s 

COS a 

B 


1st term 
2d term 


3d and 4th \ 
terms / 

— A<f> 

K<£+<£') 


4.9745303 
9. 9794727 
8. 5097191 


3.4637221 
// 

+2908.8550 
+ 2.8851 


+2911.7401 
+ 0.1299 


+2911.8700 

o t tr 

54 28 41.9 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec <£' 


JX 


9. 94906 
8. 95521 
1. 55589 


0. 46016 

tr 

+0.1972 
-0.0673 


-117 
4.9745303 
9. 4776065 
8.5087606 
0. 2315532 


3.1924389 

n 

+1557. 5388 


(^) 2 


Arg. 

s 

JX 


Corr. 


6.9283 
2.3667 


9. 2950 


-159 
+ 42 


-117 


-h 

s 2 sin 2 a 
E 


3. 4637 
8. 9043 
6. 4597 

8. 8277 


JX 

sin $(4>-\-4>') 
sec \(A<t>) 


—Act 


3.1924389 
9. 91056S7 
108 


3.1030184 

ft 

+ 1267. 70 


STATION NICHOLS 


O 


/ 

ft 

17 

28 

39.9 

+ 54 

28 

47.9 

71 

57 

27.8 

— 

40 

14.3 

180 




251 

17 

13.5 

89 

23 

18.6 

131 

21 

58. 295 

+ 

49 

14.397 

132 

11 

12. 692 




+ 1 

-h 


2. 7458 

s 2 sin 2 a 

9.4465 

E 


6. 4597 



8.6520 

JA 


3. 

4704688 

sin £(<£+<£') 

9. 9123203 

sec i(A<j>) 





3. 3827891 




ft 

—Act 

+2414.3 


Second angle 


Act 


<J> 

A<j> 

P 


s 

COS a 

B 

h 

1st term 
2d term 


3d and 4 th \ 
terms / 

-A4> 


Lazaro to Tow Hill 
Tow Hill and Nichols 

Lazaro to Nichols 


Nichols to Lazaro 


First angle of triangle 


54 


52 

9 


54 

4. 7451634 
9. 4909674 
8.5097191 


57. 843 
27. 012 


2. 7458499 

ft 

+556.9933 
+ 10.0559 


+567.0492 
- 0.0374 


+567. 0118 

O / II 

54 48 14.3 


43 

s 2 
sin 2 ! 
C 


3d term 
4th term 


s 

sin a 
A' 

sec 4>' 


JA 


30. 831 


Lazaro 

Nichols 


9.49033 
9.95620 
1.55589 


1.00242 

ft 

+0.0075 
-0.0449 


+92 
4. 7451634 
9.9781021 
8.5087447 
0. 2384494 


3. 4704688 

ft 

+2954.3966 


Arg. 

s 

JA 


Corr. 


A 

JA 


5. 5072 
2.3667 


7.8739 


- 56 
+ 148 


+ 92 





























































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


43 


primary triangulation —Continued 

STATION TOW HILL 


Third angle 


Joe 


<f> 

J4> 

V 


s 

COS a 

B 


1st term 
2d term 


3d and 4th 1 
terms / 

-J<f> 

*(*+*') 


Tower to Lazaro 
Tow Hill and Lazaro 

Tower to Tow Hill 


Tow Hill to Tower 


54 


35 

31 


27. 323 
01.351 


54 


4. 8701436 
9.8881103 
8. 5097404 


04 


25.972 


3. 2679943 

tt 

+ 1853. 5073 
+ 7.8786 


+ 1861.3859 
- 0.0351 


+ 1861.3508 

or tt 

54 19 56. 6 


*2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec <f)' 


J\ 


Tower 
Tow Hill 


9. 74029 
9. 60494 
1.55122 


0. 89645 

tt 

+0.0809 
-0.1160 


+ 16 
4.8701436 
9.8024699 
8.5087606 
0. 2315532 


3.4129289 

n 

+2587. 7893 


(<?g ) 2 


Arg. 

s 

J\ 


Corr. 


X 

J\ 


6. 5397 
2. 3683 


8.9080 


- 98 
+ 114 


+ 16 


o 



tt 

150 

-111 


32 

09 

33.9 

20.5 

39 

23 

35 

13.4 

02.4 

180 

218 

48 

11.0 

131 

+ 

04 

43 

48.045 

07. 789 

131 

47 

55. 834 

-h 

s 2 sin 2 a 
E 


3. 2680 

9. 3452 

6. 4516 

• 


9. 0648 

LP) 3 

F 


0.239 

7.733 



7. 972 

JX 

sin %(4>~\~4 > ') 
sec $(J4>) 

3. 4129289 

9. 9097770 

44 



3.3227103 




tt 

—Joe 


+2102.38 


STATION NICHOLS 







O 

t 

ft 

a 

Tow Hill to Lazaro 


197 

07 

32. 2 

Third angle 

Nichols and Lazaro 


- 36 

08 

04.3 

a 

Tow Hill to Nichols 


160 

59 

27.9 

Joe 





— 

18 

55.8 






180 



a' 

Nichols to Tow Hill 


340 

40 

32.1 


O 

t 

ft 






<t> 

54 

04 

25.972 

Tow Hill 

X 

131 

47 

55. 834 

J<f> 

+ 

39 

04. 860 


JX 

+ 

23 

16. 859 


54 

43 

30. 832 

Nichols 

X' 

132 

11 

12. 693 


s 

COS ce 

B 


1st term 
2d term 


3d and 4 th 1 
terms / 

-J<f> 


4. 8851274 
9.9756468 
8. 5097780 


3.3705522 

tt 

-2347. 2114 
+ 2.1824 


-2345.0290 
+ 0.1694 


-2344.8596 

Of ft 

54 23 58. 9 


s 2 

sin 2 a 
C 


3d term 
4 th term 


s 

sin a 
A' 

sec 4 V 


JX 


9. 77025 
9. 02568 
1. 54301 


0. 33894 

n 

+0.1292 
+0.0402 


-70 
4.8851274 
9. 5128381 
8.5087447 
0. 2384494 


3.1451526 

rt 

+ 1396.8592 






Arg. 

s 

JX 


Corr. 


6. 7403 
2.3709 


9.1112 


-104 
+ 34 


- 70 


—h 

s 2 sin 2 a 
E 


3.3706 
8. 7959 
6. 4373 

8. 6038 


JX 

sin i(0+<£ , ) 
sec i(J4>) 


—Jot 


3.1451536 
9.9101427 
70 


3. 0553033 

tt 

+ 1135.80 





































































44 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation , 

STATION KEN 


Second angle 


Ja 


4 > 

J<f> 

4 >' 

s 

COS a 

B 

h 


1st term 
2d term 


3d and 4th 1 
terms / 

-A4> 

\{<t>+4>') 


Lazaro to Nichols 
Nichols and Ken 

Lazaro to Ken 


Ken to Lazaro 


54 


+ 


52 

1 


54 

4.6073998 
8.8953917 
8. 5097191 


57. 843 
37. 056 


2.0125106 
" ( 

- 102.9226 
+ 5.8614 


- 97.0612 
+ 0.0050 


- 97.0562 

o / tt 

54 53 46.4 


54 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec <j>' 


J\ 


First angle of triangle 

Lazaro X 

JX 


34. 899 

Ken 

X' 

9. 21480 



9. 99731 

(«J£) 2 

4. 0250 

1. 55589 

2. 3667 

0. 76800 


6.3917 

+0.0002 


• 

+0.0048 



+58 



4. 6073998 



9. 9986545 

Arg. 


8. 5087403 

s 

-29 

0.2404328 

J\ 

+87 

3.3552332 

n 

+2265. 8607 

Corr. 

+58 


O 

/ 

ft 

71 

57 

27.8 

+ 22 

33 

00. 2 

94 

30 

28.0 

— 

30 

53.7 

180 



273 

59 

34.3 

116 

59 

49.5 

131 

21 

58. 295 

+ 

37 

45. 861 

131 

59 

44.156 


-h 

s 2 sin 2 a 
E 


2. 0125 
9. 2121 
6. 4597 


7. 6843 


JX 

sin 

sec 


-Ja 


3. 355234 
9. 912798 


3. 268032 

n 

+ 1853. 67 


STATION ROUND 











O 


/ 

ft 

a 

Lazaro to Ken 






94 

30 

28.0 

Second angle 

Ken and Round 






+ 78 

57 

20. 9 

a 

Lazaro to Round 






173 

27 

48.9 

Ja 









— 


i 

37. 8 










180 




a' 

Round to Lazaro 






353 

26 

11.1 







First angle of triangle 

74 

/ 42 

34. 2 

<t> 

O 

54 


/ 

52 

ft 

57. 843 


Lazaro 

A 

131 

21 

58. 295 

J<j> 

+ 


9 

58. 318 



JA 

+ 


i 

59. 402 

V 

55 


02 

56.161 


Round 

A' 

131 

23 

57. 697 

S 

4. 2700631 

s 2 


8. 54013 



-h 


2 

.7769 

COS a 

9.9971678 

sin 2 

CL 

8.11255 

GW 2 

5.5538 

s- sin 2 

a 

6. 6527 

B 

8. 5097191 

C 


1. 55589 

D 

2.3667 

E 


6.4597 

h 

2. 7769500 



8. 20857 


7.9205 



5. 8893 


ft 




ft 








1st term 

-598.3427 

3d term 

+0.0083 







2d term 

+ 0.0161 

4th term 

+0.0001 








-598.3266 











3d and 4 th 1 

+ 0.0084 

5 


-6 

4. 2700631 







LCl Hid J 



sin a 

9. 0562746 

Arg. 


JA 


2. 077013 

— J(J) 

-598.3182 

A' 


8. 5087369 

s 

-6 

sin §(<£+<£') 

9. 913183 


O / 

ft 

sec <j>' 

0. 2419389 

J\ 

0 

sec £(J<£) 




54 57 

57 



2. 0770129 

Corr. 

-6 



1.990196 




J\ 


+ 119.4024 



—Ja 


+97. 77 






























































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


45 


'primary triangulation —Continued 

STATION KEN 










0 


/ 

ft 

a 

Nichols to Lazaro 





251 

17 

13.5 

Third angle 

Ken and Lazaro 





- 40 

27 

12.5 

a 

Nichols to Ken 





210 

50 

01.0 

Aa 








+ 


9 

22.8 









180 




a’ 

Ken to Nichols 





30 

59 

23.8 

* 

O 

54 


t \ ft 

43 30.831 

Nichols 

X 

132 

11 

12. 693 

A<i> 

+ 


11 t 04.068 


. 

AX 

— 

11 

28.537 

<y 

54 


54 | 34.899 


Ken 

X' 

131 

59 

44.156 

s 

4.3790213 

S 2 

8.75804 



-h 


2. 8226 

COS a 

9.9338209 

sin 2 a 

9.41947 

(d<f>) 2 

5.6450 

s 2 sin 2 a 

8.1775 

B 

8. 5097307 

C 

1.55337 

D 

2.3676 

E 


6. 4553 

h 

2.8225729 


9. 73088 


8.0126 



7. 4554 

1st term 

-664.6192 

3d teitn 

+0. 0103 







2d term 

+ 0.5381 

4 th term 

+0.0029 








-664.0811 










3d and 4 th 1 
terms f 

+ 0.0132 

$ 

-2 

4.3790213 









sin a 

9.7097334 

Arg. 


AX 


2. S3792S 






— A4> 

-664.0679 

A' 

8.5087403 

s 

-10 

sin 

9. 912392 


O / ft 


sec y>’ 

0. 2404328 

AX 

+ 8 

sec i(A<f>) 




54 49 02. 9 


2. 8379276 

Corr. 

- 2 



2. 750320 




AX 

-688. 5375 



—Aa. 


-562. 76 


STATION ROUND 







o 

/ 

ff 

a 

Ken to Lazaro 



273 

59 

34.3 

Third angle 

Round and Lazaro 


- 26 

20 

06.8 

a 

Ken to Round 



247 

39 

27.5 

Aa 





+ 

29 

17.8 




- 


180 



a ' 

Round to Ken 



68 

08 

45.3 


O 

/ 

ft 






<t> 

54 

54 

34. 899 

Ken 

X 

131 

59 

44.156 

A<j> 

+ 

8 

21. 261 


AX 

— 

35 

46.458 

<t>' 

55 

02 

56.160 

Round 

X' 

131 

23 

57.698 




+ 1 

< 



-1 


s 

COS a 

B 

4. 6149334 
9. 5799436 
8. 5097172 

s 2 

sin 2 a 

C 

9. 22987 

9. 93222 

1. 55631 

(W 

5.4091 
2.3666 

-h 

s 2 sin 2 a 

E 

h 

2. 7045942 


0. 71840 


7. 7757 


1st term 

2d term 

-506. 5172 
+ 5.2288 

3d term 

4 th term 

+0.0058 
+0.0212 




3d and 4th \ 
terms / 

-501.2884 

+ 0.0270 

s 

sin a 

A' 

sec </>' 

+49 
4. 6149334 
9. 9661083 
8. 5087369 
0. 2419389 

Arg. 

s 

AX 


AX 

sin 

sec l(A(f>) 

— A<f> 

-501. 2614 

O / ff 

-30 
+ 79 


54 58 46.0 


3.3317224 

Corr. 

+49 




AX 

-2146.458 



—Aa 


2. 7046 
9.1621 
6. 4604 


8.3271 


3.331722 
9. 913254 


3.244976 

ff 

-1757.83 






























































40 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation, 

STATION CAT 



• 





o 

t 

tt 

a 

Lazaro to Round 



173 

27 

48.9 

Second angle 

Round and Cat 




+ 26 

22 

55.0 

Ct 

Lazaro to Cat 




199 

50 

43.9 

Ja 






+ 

4 

21.5 







180 



a' 

Cat to Lazaro 




19 

55 

05.4 

1 




1 

First angle of triangle 

89 

47 

52.'8 


O 

t 

tt 






<!> 

54 

52 

57. 843 

Lazaro 

X 

131 

21 

58. 295 

J4> 

+ 

8 

28. 257 


JX 

— 

5 

19. 363 

V 

55 

01 

26.100 

Cat 

X' 

131 

16 

38. 932 


s 

4. 2230584 

S 2 

8.44612 



—h 

cos a 

9. 9734102 

sin 2 a 

9. 06164 

(W 

5.4124 

s 2 sin 2 a 

B 

8. 5097191 

C 

1. 55589 

D 

2.3667 

E 

h 

2. 7061877 


9.06365 


7. 7791 


1st term 

-508.3791 

3d term 

+0. 0060 




2d term 

+ 0.1158 

4 th term 

+0.0005 





-508. 2633 






3d and 4th \ 

+ 0.0065 

s 

-3 

4. 2230584 




terms / 


sin a 

9. 5308211 

Arg. 


JX 

— J<f> 

-508. 2568 

A' 

8. 5087375 

s 

-5 

sin i(4>-4>') 


o t tt 

sec 4 >' 

0. 2416677 

JX 

+2 

sec l(J4>) 

X M+P) 

54 57 42.0 


2. 5042844 

Corr. 

-3 




JX 

-319.3629 



—Ja 


2. 7062 
7. 5078 
6. 4597 

6. 6737 


2. 504284 
9.913117 


2. 417*01 

tt 

-261. 46 


STATION BEAVER 








o 

t 

tt 

a 

Cat to Round 




109 

42 

58.1 

Second angle 

Round and Beaver 



+ 87 

34 

56.4 

a 

Cat to Beaver 




197 

17 

54.5 

Ja 




- 


+ 

1 

40. 2 







180 



a' 

Beaver to Gat 




17 

19 

34.7 





First angle of triangle 

49 

58 

12.0 



t 

tt 






<f> 

55 

01 

26.100 

Cat 

X 

131 

16 

38. 932 

J4> 

+ 

3 

45.192 


JX 

— 

2 

02. 273 

<t>’ 

55 

05 

11.292 

Beaver 

X' 

131 

14 

36. 659 


5 

COS a 

B 


1st term 
2d term 


3d and 4 th 
terms 

— J<J> 


3.8629808 
9.9798982 
8. 5097090 


2.3525880 

tt 

-225. 2102 
+ 0.0170 


-225.1932 
+0.0012 


-225.1920 


55 03 18. 7 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 
sec 4> 


JX 


7.7260 
8.9465 
1. 5584 


8. 2309 

tt 

+ 0.0012 


-1 

3.8629808 
9. 4732671 
8. 5087360 
0. 2423463 


2. 0873301 

u 

-122. 2728 


tfg) 2 


Arg. 

5 

JX 


Corr. 


4.705 
2.366 


7.071 


-1 


-h 

s 2 sin 2 a. 
E 


2.353 
6. 672 
6.464 


5.489 


JX 

sin i(<£+<£’) 
sec %(J<f>) 


-Ja 


2.087330 
9.913657 


2.000987 

tt 

-100.23 
































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


47 


primary triangulation —Continued 

STATION CAT 


COS a 

B 


1st term 
2d term 


3d and 4 th \ 
terms / 

— J(f> 
*(<£+<£') 







O 

f 

U 

cc 

Round to Lazaro 



353 

26 

11. 1 

Third angle 

Cat and Lazaro 



- 63 

49 

12.6 

a. 

Round to Cat 



289 

36 

58.5 

da 





+ 

5 

59.6 

a' 

Cat to Round 



180 

109 

42 

58.1 


O 

f 

n 







55 

02 

56.161 

Round 

X 

131 

23 

57. 697 

d<J> 

— 

1 

30.061 


d\ 

— 

7 

18. 765 

4>’ 

55 

01 

26.100 

Cat 

X' 

131 

16 

38.932 


3. 9177934 
9.5259756 
8. 5097071 


1. 9534761 

ft 

+89. 8413 
+ 0.2199 


+90.0612 


+90.0612 

o t n 

55 02 11.1 


sin 2 a 
C 


3d term 
4th terra 


s 

sin a 
A' 

sec <£' 


d\ 


7. 8356 
9. 9481 
1. 5586 


9. 3423 

tf 

+0. 0002 
-0.0002 


+2 

3.9177934 
9.9740335 
8. 5087375 
0. 2416677 


2. 6422323 

n 

-438. 7654 


GW 


Arg. 

s 

d\ 


Corr. 


3.9069 
2.3658 


6. 2727 


-1 

+3 


+2 


-h 

s 2 sin 2 a 
E 


1. 9535 
7. 7837 
6. 4643 


6. 2015 


d\ 

sin £(<£+<£') 
sec \{d<J>) 


—da 


2. 642232 
9.913558 


2. 555790 

ft 

-359. 58 


STATION BEAVER 







o 

f 

// 

ot 

Round to Cat 



289 

36 

58.5 

Third angle 

Beaver and Cat 



- 42 

26 

51. 8 

a 

Round to Beaver 



247 

10 

06.7 

Joe 





+ 

7 

40.0 






ISO 



a' 

Beaver to Round 



67 

17 

46.7 


o 

/ 

/f 






r}> 

55 

02 

56.161 

Round 

X 

131 

23 

57. 697 


+ 

2 

15.131 


d\ 

— 

9 

21.038 


55 

05 

11.292 

Beaver 

< X' 

131 

14 

36. 659 


s 

COS a 

B 

4.0333437 
9.5888562 
8.5097071 

s 2 

sin 2 a 

C 

8.0667 

9. 9291 

1. 5586 

(<?g) s 

4.264 
2.366 

—h 

s 2 sin 2 a 
E 

h 

2.1319070 


9. 5544 


6. 630 



// 


// 




1st term 

2d term 

-135. 4899 
+ 0.3584 

3d term 
4th term 

+0.0004 
+0. 0004 





-135.1315 


+3 

4.0333437 
9.9645661 
8.5087360 
0.2423463 




3d and 4th \ 
terms / 

+ 0.0008 

s 

sin a 

A' 

sec <f>' 

Arg. 

s 

d\ 


d\ 

sin l(<j>+'t> 
sec 

— d<{) 

-135.1307 

o f n 

-2 

+5 


55 04 03.7 

d\ 

2.7489924 

it 

-561. 0382 

Corr. 

+3 

— Joe 


2.132 
7.996 
6. 464 


6. 592 


2. 748992 
9.913723 


2.662715 
// 

-459.95 


91865°—15-4 































































48 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation, 

STATION LIM 








O 

t 

n 

CL 

Beaver to Cat 




17 

19 

34.7 

Second angle 

Cat and Lim 




+102 

39 

47.2 

a 

Beaver to Linn 




119 

59 

21.9 

Aa 






— 

5 

20.3 







180 



a' 

Lim to 

Beaver 




299 

54 

01.6 





First angle of triangle 

36 

34 

55.5 


O 

/ 

tt 






4> 

55 

05 

11.292 

Beaver 

X 

131 

14 

36. 659 

J<f> 

+ 

2 

08. 973 


AX 

+ 

6 

30. 479 

4 >’ 

55 

07 

20. 265 

Lim 

A' 

131 

21 

07.138 


5 

3.9025495 

S 2 

7. 8051 



COS a 

9. 6988310 

sin 2 a 

9. 8751 

(W 

4. 222 

B 

8. 5097044 

C 

1. 5591 

D 

2. 366 

h 

2.1110849 


9. 2393 


6. 588 

1st term 

-129.1471 

3d term 

+0.0004 



2d term 

+ 0.1735 

4 th term 

+0.0002 




-128.9736 


+ 2 



3d and 4 th 1 

i n nnna 


3.9025495 



terms / 

-f- W. UUUO 

^ s 

9.9375769 

Arg. 


— A4> 

-128.9730 

sin & 

8. 5087351 

s 

-1 


O / ft 

A 

sec 4> 

0. 2427356 

AX 

+3 

h(4>+<t>') 

55 06 15. 8 


2. 5915973 

Corr. 

+2 



AX 

+390. 4787 




—h 

s 2 sin 2 a 
E 


2. Ill 
7. 680 
6. 465 


6.256 


AX 

sin !(<£+<£') 
sec hU<t>) 


—Aa 


2. 591597 
9. 913918 


2. 505515 

tt 

+320. 27 


STATION SOUTH TWIN 











o 


r 

tt 

CL 

Beaver to Lim 






119 


59 

21.9 

Second angle 

Lim and South Twin 





+ 78 

24 

38. 4 

CL 

Beaver to South Twin 




198 

24 

00. 3 

Aa 









+ 


1 

27. 7 










180 




o' 

South Twin to Beaver 




18 

25 

28.0 


O 


/ 


tt 

First angle of t 

riangle 

60 

40 

06.2 


55 


05 

11.292 

Beaver 

X 

131 

14 

36. 659 

A<j> 

+ 


3 

04.190 



AX 

— 


1 

46. 962 

& 

55 


08 

15. 482 

South Twin 

X ' 

131 

12 

49. 697 


Fixed latitude, 15.517 




Fixed longitude, 

49. 974 

s 

3.7783859 


s 2 

7. 5568 



-h 


2. 265 

COS a 

9.9772093 

sm 2 a 

8. 9984 

(W 

4.530 

s 2 sin 2 

CL 

6. 555 

B 

8. 5097044 


C 

1. 5591 

D 

2. 366 

E 


6. 465 

h 

2. 2652996 



8.1143 


6.896 



5.285 

1st term 

-184. 2042 

3d term 

+0.0008 







2d term 

+ 0.0130 

4 th term 










-184.1912 




_i 







3d and 4th 1 



s 

3. 7783859 







terms / 

-p U. UUUo 

sin a 

9. 4992064 

Arg. 


AX 


2. 029229 

-A4> 

-184.1904 

J 


8.5087347 

s 

-1 

sin h(4>+4>’) 

9.913958 


O / 

// 

sec 4V 

0. 2429024 

AX 


sec i(A4>) 



2( < £ + 0 , ) 

55 06 43. 4 



2.0292293 

Corr. 

-1 



1.943187 




AX 

-106.9619 



—Aa 


r r 

-87. 74 


Discrepancy in latitude: 
-0.035 


Discrepancy in longitude: 
-0.277 


X 7238.24+100= 
-2.5334 


X 7238.24+100= 
-20.0499 

































































APPLICATION OP LEAST SQUARES TO TRIANGULATION 


49 


primary triangulation —Continued 

STATION LIM 







o 

1 

II 

ot 

Cat to Beaver 



197 

17 

54.5 

Third angle 

Lim and Beaver 


- 40 

45 

17.4 

a 

Cat to Lim 



156 

32 

37.1 

Ja 





— 

3 

39.9 






180 



a ' 

Lim to Cat 



336 

28 

57.2 








-.1 

<t> 

55 

01 

II 

26.100 

Cat 

X 

131 

16 

38.932 

J.f> 

+ 

5 

54.165 


J\ 

+ 

4 

28. 206 

. <y 

55 

07 

20.265 

Lim 

X' 

131 

21 

07.138 


s 

COS a 

B 


1st term 
2d term 


3d and 4 th 1 
terms j 
-J<£ 


i(<£+<£') 


4. 0770593 
9.9625415 
8.5097090 


2.5493098 

II 

-354. 2499 
+ 0.0817 


-354.1682 
+ 0.0031 


-354.1651 


55 04 23. 2 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec <f>' 


j\ 


8.1541 
9.1999 
1. 5584 


8.9124 

it 

+0.0029 
+0.0002 


4.0770593 
9.5999381 
8. 5087351 
0. 2427356 


2.4284681 

n 

+ 268. 2057 


(W 2 


Arg. 

s 

J\ 


Corr. 


5.099 
2.3.66 


7.465 


-2 

+2 


-h 

s 2 sin 2 a 
E 


2.549 
7. 354 
6.464 


6.367 


JX 

sin 2 (<^+ rf> r ') 
sec UJ<f>) 


—Jot 


2. 428468 
9.913752 


2.342220 

II 

+219.90 


STATION SOUTH TWIN 








o 

I 

II 

a 

Lim to Beaver 




299 

54 

01.6 

Third angle 

South Twin and Beaver 



- 40 

55 

15.5 

a 

Lim to South Twin 



258 

58 

46.1 

Jet 






+ 

6 

48.1 







180 



a' 

South. Twin to Lim 



79 

05 

34.2 


O 

/ 

II 






<£ 

55 

07 

20.265 

Lim 

X 

131 

21 

07.138 

J<j> 

+ 


55. 217 


J\ 

— 

8 

17.440 


55 

08 

15. 482 

South Twin 

X' 

131 

12 

49.698 








-1 

s 

3. 9531874 

v 2 7.9064 


-h 

1.744 


COS a. 

B 


1st term 
2d term 


3d and 4 th 1 
terms / 
-J<t> 


§(<£ + 0') 


9. 2813974 
8.5097018 


1.7442866 

It 

-55. 4992 
+ 0. 2 818 

-55. 2174 
+ 0.0002 


-55. 2172 

O III 

55 07 47.9 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec (/>' 


J\ 


9.9838 
1.5597 


9. 4499 

II 

+ 0.0001 
+0.0001 

+2 

3.9531874 
9. 9919164 
8.5087347 
0. 2429024 


2.6967411 

n 

-497. 4404 


(< 5 j ^) 2 


Arg. 

s 

JX 


Corr. 


3.488 
1365 


5.853 


-2 
+ 4 


+2 


s'- sin 2 a 
E 


7.890 
6. 467 

6.101 


JX 

sin 

sec RJ<£) 


—Jot 


2. 696741 
9.914053 


2. 610794 

II 

-408.12 



































































Formation of latitude and longitude condition equations 


50 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


e 

2.29 

2.37 

2.47 

2.59 

2. 74 

2. 92 

3.14 

3.41 

3. 76 


CO 

1^ 

05 

T—i 

<N 


H 

o 

00 


+°> 

00 


I- 

CO 

>0 

~P 

CO 

rH 


rH 

rH 

rH 

r-H 

rH 

rH 

rH 

rH 

rH 


Tf« 

00 

<M 

CO 

o 


00 

<N 

CO 

"0* 

o 


CO 

CO 

*p 



»o 

»o 


CO »Q 
8 CO 
tH CO 

+ I 


''o 

-©■ 

a: 1 £ 

K 

-6- 

100 

|- 74.2 

j-—120.7 

- 56.3 

- 56.3 

[- 56.3 

j~ 56.3 

|- 19.6 

|- 25.1 

)- 11.3 

(- ,4 

1 

ft* 

+ 

to <-h eo ^ »o to >o <M h >o <n n o ^ >Q <n >0 >o 

r-H rH r-1 rH <NCOCO'-rOJlO'*tCl0 1'~OI- 

1 1 + + + + IIII++IIII+ + 

••S' O <N IN H!t< to CO rH 00 rH CO 

rH rH rH rH <NC0(N'TIN'.'fC0*O>Ol^COt'- 

+__+ i i ±_^± ±^jr ±^_± j_ v _t 

-©• 1 c 

s 

X 

NH 

100 

+20.2 

+ 9.7 

-11.0 

-11.0 

-11.0 

—11.0 

-13.4 

- 4.6 

- 2.1 

-10.0 

/'"N 

O 

^ 1 
£ 

✓< 

100 

-39.2 

-18.5 

} +48.4 

+ 0.2 

- 9.8 

+ 5.3 

+ 0.1 

+ 6.8 

+ 5.1 

+ 9.8 

03 

-10+11 

-12+14 

+ 7-9 
+ 14—15 

-25+27 

-31+32 

-42+45 

-49+52 

-59+61 

-71+72 

V 

—74+75 

o 

•O- 1 * 

"©• 

100 

-47.2 

-75.5 

-81. l{ 

- 0.3 

+16.4 

- 8.9 

- 0.1 

-12.1 

- 8.7 

- 1.1 

'o 

/< H 

✓< 1 /*' 

Si 

,< 

100 

-15.4 

+ 1.1 

+ 7.4 

j—26.3 

-22.6 

-38.8 

-47.3 

- 8.8 

- 4.3 

- 3.6 


-4+6 

-2+3 

-7+9 

+ 15-16 

+25-27 

-23+25 

-28+31 

-21+22 

-39+40 

—54+55 

-63+65 

-J- 

100 

-18.6 

+ 4.6 

-12.4 

+ 44. lj 

+37.8 

+65.1 

+22.7 

+ 15.7 

+ 7.5 

+ 0.4 

ca 

1 

-2.34 

-2.30 

-5. 30 

-0.02 

+1.07 

-0.58 

-0.01 

-1. 77 

-2.84 

-1.18 

< 

+ 

<NTPiHGCt'.»0>0 0-'f<co 

O5rHCC00'T'C<INCOTt<Ttr 
© O © IN IN + + CM* IN © 

. 1 + I + + + + + + + 

1 

1$ 

/. 

+ 16.76 

+ 8.03 

- 9.14 

C 

- 9.14 

- 9.14 

- 9.14 

-11.13 

- 3.82 

- 1.78 

-8.29 

-©■ 

1 

s 

-©• 

+20. IS 

+32.84 

+ 15. 31 

+ 15.31 

+15.31 

+ 15.31 

+ 5.33 

+ 6. S3 

+ 3.08 

+ 0.92 

r< 

O / 

130 56.07 

131 04.80 

131 21.97 

131 21.97 

131 21.97 

131 21.97 

131 23.96 

131 16.65 

131 14.61 

131 21.12 

131 12.83 

“©• 

O / 

54 48.11 

54 35.46 

54 52.96 

54 52.96 

54 52.96 

54 52.96 

55 02.94 

55 01.43 

55 05.19 

55 07.34 

55 08.26 

Station 

Turn 

Tower 

Lazaro 

Lazaro 

Lazaro 

Lazaro 

Round 

Cat 

Beaver 

Lim 

South Twin 


































APPLICATION OF LEAST SQITAEES TO TETANGTTLATTON. 


51 


Latitude equation 

0= —2.5334 -0.14(2)+0.14(3)+0.39(4) —0.39(6) -0.69(7)+0.69(9)+0.67(10) —0.67(11) 
-0.66(12)- 1.36(14)+1.25(15) - 0.55(16) - 0.09(21) + 0.09(22) - 0.49(23) + 0.93(25) 
-0.44(27)- 0.76(28)+ 0.49(31)+ 0.27(32)- 0.20(39) + 0.20(40)- 0.02(42) + 0.02(45) 
+0.14(49)- 0.14(52)- 0.10(54)+ 0.10(55)+ 0.08(59)- 0.08(61) + 0.10(63) - 0.10(65) 
+0.07(71) - 0.07(72)+ 0.11(74) - 0.11(75) 


Longitude equation 

0= -20.0499+1.20(2)-1.20(3)-0.59(4)+0.59(6)+0.41(7)-0.41(9)-0.35(10)+0.35(11) 
+1.39(12) - 0.34(14) - 0.75(15)- 0.30(16)+ 0.67(21) - 0.67(22) - 0.34(23) + 0.07(25) 
+0.27(27) - 0.17(28) - 0.29(31) + 0.46(32) - 0.16(39) + 0.16(40) - 0.62(42) + 0.62(45) 
+0.20(49) - 0.20(52) - 0.07(54)+ 0.07(55) - 0.32(59) + 0.32(61) + 0.07(63) - 0.07(65) 
-0.16(71)+ 0.16(72) - 0.06(74) + 0.06(75) 


52 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 2S, 


Correlate 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

1 

-1 

-1 





















2 

+ 1 


-1 




















3 

+ 1 

+ 1 




















4 

-1 




















5 

-1 





















6 

+ 1 

+ 1 





















7 





















8 




— 1 



















9 



-1 

+ 1 



















10 

-1 


+ 1 



















11 

+ 1 





















12 

-1 

— 1 




















13 


+ 1 





















14 


+ 1 

—1 



















15 






















16 




+1 

-1 

-1 

















17 




+ 1 



-1 

-1 














18 





+ 1 















19 







+ 1 


-1 













20 










-1 











21 









+ 1 

+ 1 

-1 












22 









+ 1 

+1 











23 





-1 


-1 














24 





-1 

+1 
















25 




— 1 

+ 1 

+1 
















26 




+ 1 

















27 






















28 









-1 














29 








-1 














30 







-1 















31 





-1 


+ 1 

+ 1 














32 





+ 1 


+ 1 














33 





-1 
















34 






+ 1 

-1 
















35 






+ 1 
















36 







-1 


+ 1 













37 








+ 1 














38 









-1 













39 















-1 








40 











-1 


-1 

-1 

+ 1 








• 

41 













+ 1 








42 









-1 

-1 

+ 1 











43 











+ 1 










44 










+ 1 












45 









+ 1 













46 












-1 










47 












-1 

+ 1 










48 












+ 1 










49 







. - 




-1 

-1 











50 














-1 


-1 







51 












+ 1 

-1 










52 











+ 1 

+ 1 

+ 1 

-1 








53 













-1 






54 


















-1 





55 















+ 1 

+ 1 

+ 1 

+ 1 





56 














-1 





57 
















-1 







58 














+ 1 


+ 1 







59 














-1 

-1 

-1 

-1 





60 



* 












+ 1 





61 















+ 1 







62 
















+ 1 


-1 




63 

















+T 

+ 1 

_1 



64 



















-1 

f- ] 


65 




















+ 1 













































































































































































APPLICATION OF LEAST SQUAPtES TO TPIANGULATION 


53 


equations 


23 

24 

25 

26 

27 

28 

29 

30 

J! 

31 

l 

32 

33 

X 

34 

-0.55 
+0.47 
+0.08 








-1 
+ 1 











-0.14 
+0.14 
+ 0. 92 

-0.14 
+0.14 
+0. 39 

+ 1.20 
-1.20 
-0. 59 


































-0.92 
-4.49 

-0.39 

-0.69 

+ 0.59 
+0.41 


-4. 49 
+ 1.88 
+2. 61 
















+0.01 
+0.24 

-0. 25 
-0.43 
+ 1.14 
-0.71 







+ 1 
-1 

+4.49 
+ 2.34 

-2.34 
+2.30 

+0.69 
+0. 67 

-0. 67 
+0. 66 

-0.41 
-0.35 

+0.35 
+ 1.39 






























-5.30 
+8.18 

-2.88 







-1 

-7. 60 
+8.18 

-2. 88 

-1.36 
+ 1.25 

-0.55 

-0.34 
-0. 75 

-0.30 








-0.27 
+0. 74 
-0.47 








-0.30 


















+0. 52 












-4. 20 

+8.45 

-4.25 











-0. 22 




+ 1 

-4. 25 
+ 4.25 
-2.47 

-0.09 
+0.09 
-0. 49 

+0. 67 
-0. 67 
-0. 34 









-0. 40 
+ 1.34 
-0.94 















-0. 74 

+3. 64 
-2.90 







+5. 37 

+0.93 

+0.07 
















-2.90 

-4.25 

-0. 44 
-0. 76 

+0. 27 
-0.17 



-1.20 
+ 1.92 
















-1.41 

+ 1.25 
+0.16 











-0. 72 






+3.18 
+ 1.07 

+0.49 
+0. 27 

-0.29 
+0. 46 

















































































+0.59 
+ 1. 07 

-1.66 




-2.30 
+ 2.30 

-0.20 

+0.20 

-0.16 
+0.16 








+ 1 











-0.17 




-1 

+0.58 

-0.02 

-0. 62 











+0.65 

-0.48 

















-0. 58 

+0. 02 

+0. 62 




-3. 79 
+3.20 
+0. 59 
-0. 01 




< 































+0.01 

+0.14 

+0.20 





-4. 22 








+2. 99 
-2.98 











+ 2.73 



-1 

-0.01 

-0.14 

-0.20 





-5.11 
+6. 37 
-1.26 










-2.44 

+2.44 

-0.10 
+0.10 

-0.07 

+0.07 






+ 1. 49 


+ 1 




































-3. 99 
+9.16 

-5.17 

-1.28 


-1 

+ 1.77 

+0.08 

-0.32 















-1. 77 

-0.08 

+0.32 






+3.47 
-2.19 









-3. 92 
+5.68 

-1.76 

+ 1 

-0. 43 

+0.10 

+0. 07 















+6. 43 

-o.io 

-0.07 


- 3.55 
+ 2.39 
+ 1.16 

- 0.28 

- 1.00 

+ 1.28 

- 9.26 
+ 0.88 
+ 8.39 
+ 1.90 

- 1.91 
+ 1.92 
+ 2.14 
-16.31 
+ 16. 86 

- 7.88 

- 0.56 
+ 0.53 
+ 0.52 

- 5.20 


6.56 
1.42 
5. 70 
1.34 
5. 69 


+ 4.64 

- 5.97 

- 7.38 
+ 0.92 

- 2.41 

+ 4.91 
+ 3.96 

- 1.00 
0.00 

+ 1.00 

0.00 
+ 1.00 
- 1.00 

- 3.07 
+ 2.73 

- 0.66 

- 2.23 
+ 1.00 
+ 1.65 
+ 0.58 

- 4.79 
+ 3.20 
+ 1.59 

- 1.66 
- 6.22 


+ 


2.99 
+ 0.40 
- 6.11 
+ 2. 76 
+ 7.84 

- 1.00 
- 1.00 
+ 2.00 
8. 74 
+ 10.16 

- 5.70 
+ 3.47 

- 4.37 
+ 4.68 
+ 0.50 


6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

61 

62 

63 

64 

65 
























































































































































54 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Correlate 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

IS 

19 

20 

21 

22 



















-1 




















-1 


+ 1 




















+ 1 



























-1 




















-1 


+ 1 


















-1 

-1 

+ 1 




















+ 1 























+ 1 























-1 

-1 





















+ 1 


-1 





















+ 1 

+ 1 





















-1 

-1 





















+ 1 






















+ 1 























66 

6-7 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 






\ 


y 




















































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


equal io ns —Continued 


23 

24 

25 

26 

» 

28 

29 

30 

a 

31 

e 

32 

4 

33 

X 

34 

1 

V 













- 1.00 
0.00 
+ 1.00 

- 3.76 
+ 3.30 

-0.03 
+ 3. 2o 

- 2.76 

- 0.62 
+ 0.95 

- 0.33 

- 1.82 
+ 1.00 
+ 0.82 
































-1.61 
+ 1.15 

+0.46 


-1.15 
+ 1.15 

+2. 84 
-2. 84 










+i 

-i 









-1.24 
+5.00 
-3. 76 

+0.07 
-0.07 

-0.16 
+0.16 





















+0.15 
+3.18 

-3.33 


+ 1.18 
-1.18 

+0.11 
-0.11 

-0. 06 
+0.06 








-i 

+i 















+0.18 























-0.18 




1 










List of corrections 



Ps.* 

Adopted 

Ps. 

P. 


Ps.* 

Adopted 

Ps. 

P. 

1 

+0.699 

+0.7 

0.49 

41 

-0. 789 

-0.8 

0.64 

2 

+2. 448 

+2.4 

5.76 

42 

-1.997 

-2.0 

4.00 

3 

-3.146 

-3.2 

10.24 

43 

+0. 028 

+0.0 

0.00 

4 

-1.369 

-1.4 

1.96 

44 

+ 1.554 

+ 1.6 

2.56 

5 

-0.207 

-0.2 

0.04 

45 

+0.581 

+0.6 

0.36 

6 

+ 1.576 

+ 1.6 

2. 56 

46 

+0.697 

+0.7 

0. 49 

n 

i 

+0. 806 

+0.7 

0.49 

47 

-1.478 

-1.5 

2.25 

8 

-1.876 

-1.9 

3.61 

48 

+0.781 

+0.8 

0.64 

9 

+0. 498 

+0.5 

0.25 

49 

+0. 735 

+0.8 

0. 64 

10 

-0.117 

-0.1 

0.01 

50 

+ 1.145 

+ 1.2 

1.44 

11 

+0.688 

+0.7 

0. 49 

51 

+0.294 

+0.3 

0.09 

12 

+3.097 

+3.1 

9.61 

52 

-0.317 

-0.3 

0.09 

13 

+ 1.159 

+ 1.2 

1.44 

53 

-1.522 

.—1.5 

2.25 

14 

-2.691 

-2.7 

7.29 

54 

-0.138 

-0.1 

0.01 

15 

-1.472 

-1.4 

1.96 

55 

-0.197 

-0.2 

0.04 

16 

-0. 728 

-0.7 

0.49 

56 

+0. 741 

+0.7 

0.49 

17 

+0. 755 

+0.8 

0. 64 

57 

+0.525 

+0.5 

0.25 

18 

-0.168 

-0.1 

0.01 

58 

-1.266 

-1.3 

1.69 

19 

+0. 945 

+ 1.0 

1.00 

59 

-0. 592 

-0.6 

0.36 

20 

-0.090 

-0.1 

0.01 

60 

-0. 262 

-0.2 

0.04 

21 

+0.102 

+0.1 

0.01 

61 

+0.524 

+0.6 

0.36 

22 

-0. 910 

-0.9 

0.81 

62 

+ 1.193 

+ 1.2 

1.44 

23 

-1.665 

-1.6 

2.56 

63 

+0.065 

+0.1 

0.01 

24 

+0. 614 

+0.7 

0. 49 

64 

+0.364 

+0.4 

0.16 

25 

-1.570 

-1.5 

2.25 

65 

-1.294 

-1.3 

1.69 

26 

+2.090 

+2.1 

4.41 

66 

-0.190 

-0.2 

0.04 

27 

+0.530 

+0.5 

0.25 

67 

-0. 898 

-0.9 

0. 81 

28 

-0. 966 

-1.0 

1.00 

68 

+ 1.088 

+ 1.1 

1.21 

29 

+0.1&3 

+0.2 

0.04 

69 

-0. 748 

-0.8 

0.64 

30 

-1.164 

-1.2 

1.44 

70 

-0.490 

-0.5 

0.25 

31 

+0.311 

+0.3 

0.09 

71 

. +0.134 

+0.1 

0.01 

32 

+ 1.636 

+ 1.6 

2.56 

72 

+ 1.234 

+ 1.2 

1.44 

33 

-0. 457 

-0.4 

0.16 

73 

-0.130 

-0.2 

0.04 

34 

+ 1.167 

+ 1.2 

1.44 

74 

+ 1.360 

+1.3 

1.69 

35 

-0. 710 

-0.7 

0. 49 

75 

-0.203 

-0.2 

0.04 

36 

-0. 494 

-0.5 

0.25 

76 

-1.158 

-1.2 

1.44 

37 

+ 1.484 

+ 1.5 

2.25 

77 

+0.445 

+0.4 

0.16 

38 

-0. 990 

-1.0 

1.00 

78 

-1.482 

-1.5 

2.25 

39 

40 

-0.319 

+0.941 

-0.3 

+0.9 

0.09 

0.81 

79 

+ 1.037 

Total. 

+ 1.0 

1.00 

103.76 


55 

66 

67 

68 

69 

70 

71 

72 

73 

74 

75 

76 

77 

78 

79 


* These values result from the computation on p. 69, 



































































Normal equations 


50 COAST AND GEODETIC SURVEY SPECIAE PUBLICATION NO. 28. 


* 

C/3 

o 

+0.0070 

+0.2070 

-0. 7922 

+1.9492 

+ 1.1901 

+0. 4571 

-0. 7099 

+ 1.4844 

-0.3254 

+0.9897 

CO 

b 

-1.2334 

+0.8942 

+0.0285 

-0. 7413 

-0.0027 

-0.5247 

+1.0884 

+0.5038 

+0.1899 

-0.0433 

'N 

- 1.81 
+ 11.21 
-13.95 
+ 6.89 
+14. 86 

+13.76 

+ 11. 21 

+ 3.77 

+19.02 

+ 7.72 


- 3.04 

+ 8.66 

+ 7.37 

+ 11.43 

+13. 78 

+38. 76 

+22.06 

+11.03 

- 7.47 

+ 5.41 

R* 

>000001 O<MC0<N<M 
io + <n'coc4 oeooicoco 

1 -f + 1 1 1 1 1 1 1 

R* 

O O N CM >C 00 rH rH CO 

id r4 cd id cm cm id 1-4 cm cm 
+ 1 ++ 1 +11++ 

CO 

OOONrfCO NOOOOJ 

O O O TT CO 00 CM CM 

CO C8 H d H © © © rH rH 
+111+ ++ 1 ++ 

Tji 

CO 

CM N CO O CO © © iO CO GO 
iQ GO CO CO CM CO CO 00 CM rH 

CM O O O rH OOOOO 

1111+ ++++ 1 

& 

COi-tCOt^^O 00CD05ON 

CM © *0 U- 

CM o t-H o T-H 1-4 o O rH o 

111 ++ ++++ 1 

3 

CM lO ^-rf 00 CM CM 00 CO iO 

co o co co th ooooco 
ddddo o o o o o’ 
1111+ ++++ 1 

CM 

CO 

CO GO Tf rH lO^OOCMCO 

O O N 00 CO CM *0 H O GO 

cd cd rH cd go oocdcdcM + 

1 M ++ ++++ I 

CM 

CO 

CO^f HHH U- U- ONO 

CM CO CO iO CO CO O CM rH 

cd + cm cm cd o o cd cd o 
++ii+ ++ii+ 

rH 

CO 

CO rH rf CM CM CM 

++1+ ++ 

rH 

CO 

CM CM ’rf CM CM CM 

1 11+ ++++1 

t>* 

CM 

+8.45 
+8.45 

8 

-4.38 
-4.38 
+4.50 

CO 

cm 

+0.42 

-1.82 

+0.25 

+0.08 

8 

+0.02 

+0.02 
+8. 60 
-2.30 
-8.18 
+0.95 

a 

+0.67 

-0.62 

-2.48 
+3.31 
+0.51 
+0.51 

00 

CM 

+ 1.66 

+ 1.66 
+ 4.22 
- 1.94 

+18.86 
+ 5.48 
+ 5.48 

CM 

-7.91 
+ 7.53 
+ 2.14 

+2.14 

CM 

!>• Tf CM 00 00 

CO CO O C5 o 

id O rH CM CM 

7++ i + 

CO 

CM 

CO O H CM 
l O CM Tf N 

© CM* © © 

+ + 1 + 

CO 

CM 

+0.05 

rH 

tH 

CM CM 

1 1 

CM 

CM 

CM 

1 

o 

rH 

CM CM CO 

1 + + 

rH 

CM 

CM 

+ 

CT> 

CM CM CO 

1 + + 

O 

CM 

CM CM CO 

1 1 + 

00 

CM CO 

1 + 

© 

H 

CM CM CO 
■ + + 


CM CM CO 

+ 1 + 

00 

rH 

<N <M <N O 
+ + + + 

CO 

CM CM CO 

1 + + 

rH 

CM CM CO 
+ + + 

10 

CM CO 

1 + 

CO 

rH 

CM CM CO 
+ + + 


CM CO 

1 + 

1C 

rH 

CM CM CM CO 

1 11 + 

CO 

CM CM CO 

1 + + 

3 

CM CM CO 
+ + + 

CM 

CM CO 
+ + 

CO 

rH 

CM CM CO 
+ 1 + 

H 

9+ 

CM 

rH 

CM CO 
+ + 


• iiii * • • • « 

• • • • • 

• « • • • • • • • • 

• • • i • (•••* 

• • • • • i • • • * 

• * • • • i i • i • 

• (••I * * * • • 

• •••• * * * * * 

• •••• * * * « • 

• * • • • « • • • • 

• •••• i«iii 

• • • • • ***** 

• • * • • • * * * • 

* • • • » • • • • • 

• • • • • ••*•• 

• • • • • 

• *••• ••*•• 

* • • • • * * * * * 

it*** •*••* 

• •••• •*••* 

• ■(it •••** 

• iiii • i * * * 

rH CM CO iO «ONCCOO 

rH 

rH 

rH 

9+ 


• * * * * * ■ * * * 

• •it* ••••* 

• •••• * * * * i 

• • • • • ii**i 

iiii* • • ■ • • 

• * t • • • • • • • 

• • • • • i * * * * 

• • • • • * * * * * 

• • • • • • i • • • 

• • • • • • • • t * 

• • • • * * * * * * 

* • • • * * * * * * 

i • • * * * * * • • 

• • • • • •••:(* 

***** * * * * * 

• lit* 1 * 1 • • 

« i « • • • • • 1 l 

i—tCMeo^io corrode*© 

HHHHH H rH rH rH CM 
























































































APPLICATION OF LEAST SQUARES TO TRIANOULATION. 


57 




§3 


3 


r-H r-H »0 OC 1 03 

OO^COCOCl 

'tOOOOOO 

r-H t-H o O r-H 

I + + + + 


o *0 »C O 

COI-QI- 
00 r-H GO »0 CO 
CO O) X' 05 
00 r-H o O rH 

0 0*000 

+ I + + I 


r-H O O CM 

I +1 + 


H 03 CO 
O CO o 
CONcOI^N 
OOhtPCN 

id oo id cd + 

+++++ 


CO 03 CO H »0 
iO CO 00 T 
»o CO CO 00 o 
COCOINOJ^ 




OHIO 
NCOhO 
X03 03C0 


+++1 


CM p 
HHINNO 

co 6 d d d 

++1 I I 


CM O CO 00 00 


CO w»oo 

n »o oi d 


I++I I I I I I 


OJOON 

CO 

H CO O ^ CO 
© O O CM r-t 

© © © © 

I I I I I 


CM © iO CO © 
CO CO CM O CO 

i'- o rr cm 

OHNNO 

©03©©© 

++1 + I 


CM r-H CM 
CM CO 
03 00 CO »C 
COOO'T 

CO CO r-H 03 

+7 i + 


Tf 00 o 

CM CO CO 
r-H r-H CO O CM 
CM H 05 »0 r-H 


00 CM 00 CM r-H 
CM CM r-H 
1^ CM HT CM r—» 
lO O lO lO 

O* O O r-I O 


I++++ +1111 


r-H 00 

CO 1-0 
NHtO 
CO CO 

r-H 03 00 
'*r 

+++ 


© © © 
CO CM ^ 
CO CM Hr O O 
03 r-H lO 00 00 



I++++ +1+1+ ++ 


10 03 

HCOHN o’ H CO O 03 CO 

++++ I++I I + 


T CO 

rH !■« 

oo 


++ 


COIN 

<N -T 


++ 


-f 50 
>0 -*r 
CO CO 

H <N 


I + 


8i 

g; 


I + 


50 

+ 


t—11— 

++ 


05 O 
00 1-H 
1^0 


++ 


IM 

+ 


<N 50 

++ 


50 

+ 


A 

a 

o 

a 

o 


3 

a 

s 

o 

o 

« 

A 

































58 COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 28, 


Solution 


of 


1 

2 

3 

23 

4 

6 

24 

7 

5 

+0 

+2 

-2 

+0.53 







C, 

-0.33333 

+0.33333 

-0.08833 








+6 

+2 

+2.20 







1 

-0.6667 

+0.6667 

-0.1767 








+5.3333 

+2.6667 

+2.0233 








c 2 

-0.50001 

-0.37937 









+6 

-0. 44 

-2 


_ 

7.91 




1 

-0.6667 

+0.1767 








2 

-1.3333 

-1.0117 









+ 4 

-1.275 

-2 


_ 

7.91 





c 3 

+0.31875 

+0.5 


+ 

1.9775 






+2.6386 

+0.72 


+ 

3.7891 





1 

-0.0468 









2 

-0. 7676 





* 




3 

-0. 4064 

-0.6375 


— 

2.5213 






+ 1.4178 

+0.0825 


+ 

1.2678 






Cm 

-0.05819 

1 

— 

0. 89420 







+6 

1 

-2 

+ 

7.53 


-2 




3 

-1 


— 

3.955 






23 

-0.0048 


- 

0.0738 







+ 4.9952 

-2 

+ 

3.5012 


-2 





c< 

+0.40038 

— 

0. 70091 


+0.40038 






+6 

+ 

2.14 

-2 

+2 





4 

-0.8008 

+ 

1.4018 


-0.8008 






+5.1992 

+ 

3.5418 

-2 

+ 1.1992 






c 6 

— 

0.68122 

+0.384675 

-0.23065 



* 




+ 156.0106 


+2.14 






3 

— 

15.6420 








23 

— 

1.1337 








4 

— 

2.4540 


+ 1.4018 






6 

— 

2.4127 

+ 1.3624 

-0.8169 







+ 134.3682 

+1.3624 

+2.7249 








C 24 

-0.0101393 

-0.0202794 









+6 

+ 2 








6 

-0.7693 

+0.4613 








24 

-0.0138 

-0.0276 









+5.2169 

+2.4337 









Ci 

- 

-0.46650 






























APPLICATION OF LEAST SQUARES TO TRIANGULATION 


59 


normals 


25 

31 

32 

33 

34 

i) 

I 

i 

+3 


6.66 

- 2.26 

+3.08 

- 5.5 


1.81 


-0.5 

+ 

1.11 

+ 0.37667 

-0.51333 

+ 0.91667 

+ 

0.30167 


+ 1 

_ 

3.08 

- 0.91 

-2.00 

+ 4.0 

+ 

11.21 


-1 

+ 

2.22 

+ 0.7533 

-1.0267 

+ 1.8333 

+ 

0.6033 



_ 

0.86 

- 0.1567 

-3.0267 

+ 5.8333 

+ 

11.8132 



+ 

0.16125 

+ 0.02938 

+0.56751 

- 1.09375 

— 

2.21499 


-4 

_ 

11.77 

- 1.76 

-4.07 

+ 12.0 

_ 

13.95 


+ 1 

— 

2.22 

- 0.7533 

+ 1.0267 

- 1.8333 

— 

0.6033 



+ 

0.43 

+ 0.0783 

+ 1.5133 

- 2.9167 

— 

5.9067 


-3 

_ 

13.56 

- 2.435 

-1.53 

+ 7.25 

_ 

20.4600 


+0.75 

+ 

3.39 

+ 0.60875 

+0.3825 

- 1.8125 

+ 

5.115 


+ 1.50 

+ 

5.5439 

+ 0.9624 

-0.0639 

- 2.22 

+ 

15.1601 


-0.2650 

+ 

0.5883 

+ 0.1996 

-0.2721 

+ 0.4858 

+ 

0.1599 



+ 

0.3263 

+ 0.0594 

+ 1.1482 

- 2.2130 

— 

4.4816 


-0.9562 

— 

4.3222 

- 0.7762 

-0.4877 

+ 2.3109 

— 

6.5216 


+0.2788 

+ 

2.1363 

+ 0.4452 

+0.3245 

- 1.6363 

+ 

4.3166 


-0.19664 

— 

1.50677 

- 0.31401 

-0.22888 

+ 1.15411 

— 

3.04458 

+0.67 

+2 

+ 

3.84 

+ 0.57 

-0.44 

- 8.0 

+ 

6.89 


-1.5 

— 

6.78 

- 1.2175 

-0. 7650 

+ 3.6250 

— 

10.23 


-0.0162 

— 

0.1243 

- 0.0259 

-0.0189 

+ 0.0952 

— 

0.2512 

+0.67 

+0.4838 

_ 

3.0643 

- 0.6734 

-1.2239 

- 4.2798 

— 

3.5912 

-0.13413 

-0.09685 

+ 

0.61345 

+ 0.13481 

+0.24502 

+ 0.85678 

+ 0.71893 

-2.48 


+ 

8.25 

+ 1.48 

+0.37 

- 0.0 

+ 

13.76 

+0.2683 

+0.1937 


1.2269 

- 0.2697 

-0.4900 

- 1.7135 

— 

1.4378 

-2.2117 

+0.1937 

+ 

7.0231 

+ 1.2103 

-0.12 

- 1.7135 

+ 

12.3221 

+0.42539 

-0.03726 

— 

1.35080 

- 0.23279 

+0.02308 

+ 0.32957 

— 

2.37000 

+ 1.4732 

+7.91 

+ 151.8020 

+24.5038 

-7.2148 

- 5.7 

+336. 4739 


-5.9325 

— 

26.8149 

- 4.8152 

-3.0256 

+ 14.3369 

— 

40.4597 


-0.2493 

_ 

1.9103 

- 0.3981 

-0. 2902 

+ 1.4632 

— 

3.8599 

-0.4696 

-0.3391 • 

+ 

2.1478 

+ 0.4720 

+0.8578 

+ 2.9998 

+ 

2.5171 

+ 1.5067 

-0.1320 


4. 7843 

- 0.8245 

+0.0817 

+ 1.1673 

— 

8.3941 

+2.5103 

+ 1.2571 

+ 120.4403 

+ 18.9380 

-9.5911 

+ 14.2672 

+286.2773 

-0.0186822 

-0.0093556 

— 

0.8963453 

- 0.1409411 

+0.0713792 

- 0.1061799 

— 

2.1305435 

+3.31 


+ 

3.54 

+ 0.76 

+0.80 

- 3.2 

+ 

11.21 

-0. 8508 

+0.0745 

+ 

2. 7016 

+ 0.4656 

-0.0462 

- 0.6591 

+ 

4.7400 

-0.0255 

-0.0127 


1.2212 

- 0.1920 

+0.0972 

- 0.1447 

— 

2.9027 

+2.4337 

+0.0618 

+ 

5.0204 

+ 1.0336 

+0.8510 

- 4.0038 

+ 

13.0473 

-0.46650 

-0.01185 


0.96233 

- 0.19813 

-0.16312 

+ 0.76747 

_ 

1 

2.50097 
































60 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Solution of 



5 

25 

8 

9 

26 

10 

12 

27 


+6 

-0.62 

-2 

-2 

+0.42 




4 

-0.8008 

+0. 2683 







6 

-0. 2766 

+0.5101 







24 

-0.0553 

-0.0509 







7 

-1.1353 

-1.1353 








+3.7320 

-1.0278 

-2 

-2 

+0.42 





a 

+0.27540 

+0.53591 

+0.53591 

-0.11254 






+7.2568 

+0.51 

+0.51 

-1.1220 





4 

-0. 0899 








6 

-0. 9408 








24 

-0. 0469 








7 

-1.1353 








5 

-0.2831 

-0.5508 

-0.5508 

+0.1157 






+ 4.7608 

-0.0408 

-0.0408 

-1.0063 






Cx 

+0.00857 

+0.00857 

+0.21137 







+6 

+2 

-1.82 

_2 





5 

-1.0718 

-1.0718 

+0. 2251 






25 

-0.0003 

-0.0003 

-0.0086 







+4.9279 

+0.9279 

-1.6035 

-2 






c 8 

-0.18830 

+0.32539 

+0.40585 







+6 

+0.25 

+2 


+8.45 




5 

-1.0718 

+0.2251 







25 

-0.0003 

-0.0086 







8 

-0.1747 

+0.3019 

+0.3766 







+4.7532 

+0. 7684 

+2.3766 


+8.45 





' c 9 

-0.16166 

-0.5 


-1.77775 






+6. 7354 

+0.08 


-1.8590 





5 

-0. 0473 








25 

-0.2127 








8 

-0.5218 

-0.6508 







9 

-0.1242 

-0.3842 


-1.3660 






+5.8294 

-0.9550 


-3.2250 






Ci6 

+0.16382 


+0.55323 







+ 6 


+8.45 






8 

-0.8117 








9 

-1.1S83 


-4.2250 






26 

-0.1564 


-0.5283 







+3.8436 


+3.6967 







Cio 


-0.96178 








+6 

+0.34 

1 







Cu 

-0.05667 






























APPLICATION OF LEAST SQUARES TO TRIANGULATION 


61 


normals —Continued 


11 

13 

31 

32 

33 

34 

V 

2 



+0.1937 
-0.0447 
-0.0255 
-0.0288 

+8.61 
-1.2269 
-1.6199 
-2.4425 
-2.3420 

+ 1.75 
-0. 2697 
-0. 2792 
-0.3841 
-0.4822 

+ 1.46 
-0. 4900 
+0.0277 
+0.1945 
-0.3970 

-2.9 
-1.7135 
+0.3952 
-0.2893 
+ 1.8678 

+14.86 

- 1.4378 

- 2.8421 

- 5.8055 

- 6.0866 



+0.0947 
-0.02538 

+0.9787 
-0.26225 

+0.3348 

-0.08971 

+0.7952 
-0.21308 

-2.6398 
+0. 70734 

- 1.3122 
+ 0.35161 



-0.0649 
+0.0824 
-0.0235 
-0.0288 
+0.0261 

+0.8640 
+0.4110 
+2.9876 
-2.2501 
-2.3420 
+0.2695 

+0.1260 
+0.0903 
+0.5148 
-0.3538 
-0.4822 
+0.0922 

-0.1377 
+0.1642 
-0.0510 
+0.1792 
-0.3970 
+0.2190 

-6.09 
+0.5740 
-0. 7289 
-0. 2665 
+ 1.8678 
-0.7270 

+ 4.2703 
+ 0.4817 
+ 5.2417 

- 5.3483 

- 6.0866 
- 0.3614 



-0.0087 

+0.00183 

-0.0600 

+0.01260 

-0.0127 
+0.00267 

-0.0233 
+0.00489 

-5.3706 
+ 1.12809 

- 1.8024 
+ 0.37859 



+0.0508 

-0.0001 

+3.18 

+0.5245 

-0.0005 

+0. 49 
+0.1794 
-0.0001 

-0.29 
+0.4262 
-0.0002 

-2.3 
-1.4147 
-0.0460 

+ 3.77 

- 0.7032 

- 0.0154 



+0.0507 
-0.01029 

+3.7040 
-0.75164 

+0.6693 
-0.13582 

+0.1360 
-0.02760 

—3. 7607 
+0. 76314 

+ 3.0516 
- 0.61925 

-2 


+2 

+0.0508 
-0.0001 
-0.0095 

+2.02 
+0.5245 
-0.0005 
-0.6975 

+1.20 
+0.1794 
-0.0001 
-0.1260 

+ 1.79 
+ 0. 4262 
-0.0002 
-0.0256 

-3.2 
-1. 4147 
-0.0460 
+0. 7081 

+ 19.02 

- 0.7032 

- 0.0154 

- 0.5746 

-2 

+0.42077 


+2.0412 
-0.42944 

+ 1.8465 
-0.38848 

+ 1.2533 
-0.263675 

+2.1904 
-0.46083 

-3.9526 
+0.83157 

+ 17. 7270 
- 3.72949 

+0.05 

+0.3233 


-0.05 
-0.0107 
-0. 0018 
+0.0165 
-0.3300 

+3.9252 
-0.1101 
-0.0127 
+ 1.2052 
-0.2985 

+0.5728 
-0.0377 
-0. 0027 
+0.2178 
-0. 2026 

+0.0732 
-0.0895 
-0.0049 
+0.0443 
-0.3541 

-2.60 
+0. 2971 
-1.1352 
-1.2237 
+0.6390 

+ 4.6556 
+ 0.1477 

- 0.3810 
+ 0.9930 

- 2.8657 

+0.3733 
-0.06404 


-0.3760 

+0.06450 

+4.7091 
-0.80782 

+0.5476 
-0.09394 

-0.3310 

+0.05678 

-4.0228 

+0.69009 

+ 2.5496 
- 0.43737 

-2 

+ 1 

+0.0612 


+2 

+0.0206 
-1.0206 
-0.0616 

-4.83 
+ 1.5033 
-0.9233 
+0. 7714 

-0.07 
+0.2716 
-0.6266 
+0.0897 

+1.29 
+0.0552 
-1.0952 
-0.0542 

-3.2 
-1.5263 
+ 1.9763 
-0.6590 

+ 7.72 
+ 1.2385 
- 8.8635 
+ 0.4177 

-0.9388 
+0.24425 


+0.9384 
-0.24415 

-3.4786 
+0.90504 

-0.3353 
+0.08724 

+0.1956 
-0.05089 

-3.4090 
+0.88693 

+ 0.5126 
- 0.13336 

+2 

-0.33333 

-2 

+0.33333 


+4.24 
-0.70667 

-0.05 
+0.00833 

-0. 87 
+0.145 

-1.0 
+0.16667 

+ 8.66 
- 1.44333 


























62 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Solution of 



27 

11 

13 

14 

15 

28 

16 

17 


+ 149. 8778 

-15. 67 

+1.02 

-2.98 

+2.98 

_ 

8.1354 



9 

- 15.0220 

+ 3.5555 








26 

- 1.7842 

+ 0. 2065 








10 

- 3.5554 

+ 0.9029 








12 

- 0.0193 

- 0.1133 

+0.1133 








+ 129.4969 

-11.1184 

+1.1333 

-2.98 

+2.98 

_ 

8.1354 




Cn 

+ 0.0858584 

-0.0087516 

+0.0230121 

-0.0230121 

+ 

0.0628231 





+ 6 

+2 

+2 

-2 

+ 

1.66 




9 

- 0.8415 









26 

- 0.0239 









10 

- 0. 2293 









12 

- 0.6667 

+0. 6667 








27 

- 0.9546 

+0.0973 

-0.2559 

+0. 2559 

— 

0. 6985 





+ 3.2840 

+2. 7640 

+ 1.7441 

-1.7441 

+ 

0.9615 





c a 

-0.84166 

-0. 53109 

+0.53109 

— 

0. 29278 






+6 

+2 

-2 

+ 

1.66 





12 

-0. 6667 









27 

-0.0099 

+0.0261 

-0.0261 

+ 

0.0712 





11 

-2.3263 

-1.4679 

+ 1.4679 

— 

0.8093 






+2.9971 

+0.5582 

-0.5582 

• + 

0.9219 






Cl3 

-0.18625 

+0.18625 

— 

0.30760 







+6 

-2 

+ 

4.22 

+ 2 





27 

-0.0686 

+0.0686 

— 

0.1872 






11 

-0. 9263 

+0. 9263 

— 

0. 5106 






13 

-0.1040 

+0.1040 

— 

0.1717 







+4.9011 

-0.9011 

+ 

3.3505 

+ 2 






Cl 4 

+0.18386 

— 

0. 68362 

- 0.40807 







+6 

— 

1.94 

+ 2 

+2 





27 

-0. 0686 

+ 

0.1872 







11 

-0. 9263 

+ 

0.5106 







13 

-0.1040 

+ 

0.1717 







14 

-0.1657 

+ 

0. 6160 

+ 0.3677 







+4. 7354 

— 

0. 4545 

+ 2.3677 

+2 






Cl5 

+ 

0.09598 

- 0.5 

-0. 42235 







+ 158. 2846 

+18.86 

+5. 48 






27 

— 

0. 5111 








11 

— 

0. 2815 








13 

— 

0.2836 








14 

— 

2. 2905 

- 1.3672 







15 

— 

0.0436 

+ 0.2273 

+0.1920 







+ 154.8743 

+ 17. 7201 

+5. 6720 








c 28 

- 0.114416 

-0.036623 









+ 6 

+2 








14 

- 0.8161 









15 

- 1.1839 

-1 








28 

- 2.0275 

-0. 6490 









+ 1.9725 

+0. 3510 




1 





Cl 6 

-0.17795 










































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


63 


normals —Continued 


18 

29 

31 

32 

33 

34 

V 

V 



+ 

11.43 

-53.9453 

-0. 7272 

+9.1030 

+0.0 

+ 108.3339 



— 

3. 6287 

- 3.2826 

-2. 2281 

-3.8940 

+7.0267 

_ 

31.5142 



— 

0. 2080 

+ 2.6052 

+0.3029 

-0.1831 

-2. 2255 

+ 

1.4105 



— 

0.9025 

+ 3.3456 

+0. 3225 

-0.1881 

+3. 2787 


0 4930 





- 0.2403 

+0.0028 

+0.0493 

+0.0567 

— 

0. 4908 



+ 

6.6908 

-51.5174 

-2.3271 

+4. 8871 

+8.1366 

+ 

77. 2464 



— 

0.0516676 

+ 0.3978273 

+0.0178703 

-0.0377391 

-0.0628324 


0.5965116 



— 

4 

+ 6.76 

-0.32 

-2.52 

+5.0 


3.04 



+ 

0.8589 

+ 0. 7770 

+0. 5274 

+0. 9217 

-1.6631 

+ 

7. 4590 



+ 

0. 0241 

- 0.3016 

-0.0351 

+0. 0212 

+0. 2576 


0.1633 



+ 

0. 229? 

- 0.8496 

-0.0819 

+0. 0478 

-0.8326 

+ 

0.1252 





- 1.4133 

+0.0167 

+0. 2900 

+0. 3333 


2. 8866 



+ 

0. 5745 

- 4.4232 

-0.1998 

+0. 4196 

+0.6986 

+ 

6.6323 



— 

2.3133 

+ 0.5493 

-0.0927 

-0.8197 

+3. 7938 

+ 

8.1269 



+ 

0. 70442 

- 0.16727 

+0.02823 

+0. 24960 

-1.15524 


2. 47470 



— 

2 

- 2.31 

-0.34 

-0.36 

+3.7 

+ 

7.37 





+ 1.4133 

-0.0167 

-0. 2900 

-0.3333 

+ 

2. 8866 



— 

0.0586 

+ 0.4509 

+0.0204 

-0. 0428 

-0.0712 


0. 6760 



+ 

1.9470 

- 0.4623 

+0.0780 

+0.6899 

-3.1931 

— 

6.8401 



— 

0.1116 

- 0.9081 

-0. 2583 

-0. 0029 

+0.1024 

+ 

2. 7405 



+ 0.03724 

+ 0.30299 

+0.08618 

+0.00097 

-0.03417 


0. 91438 



— 

2 

- 2.31 

-0. 34 

-0.36 

+5.2 

+ 

11.43 



+ 

0.1540 

- 1. 1855 

-0.0536 

+0.1125 

+0.1872 

+ 

1. 7776 



+ 

1. 2286 

- 0.2917 

+0.0492 

+0.4353 

-2.0148 


4. 3161 



+ 0.0208 

+ 0.1691 

+0.0481 

+0.0005 

-0.0191 

— 

0. 5104 



— 

0. 5966 

- 3.6181 

-0. 2963 

+0.1883 

+3.3533 

+ 

8.3811 



+ 

0.12173 

+ 0.73822 

+0.06046 

-0.03842 

-0.68419 


1. 71004 

+2 

+0.02 

+ 

4 

+ 3.51 

+0.48 

+ 1.23 

-2.5 

+ 

13. 78 



— 

0.1540 

+ 1.1855 

+0.0536 

-0.1125 

-0.1872 


1. 7776 



— 

1. 2286 

+ 0.2917 

-0.0492 

-0. 4353 

+2. 0148 

+ 

4.3161 



— 

0. 0208 

- 0.1691 

-0. 0481 

-0. 0005 

+0.0191 

+ 

0. 5104 



— 

0.1097 

- 0.6652 

-0.0545 

+0.0346 

+0. 6165 

+ 

1. 5409 

+2 

+0. 02 

+ 

2.4869 

+ 4.1529 

+0.3818 

+0. 7163 

-0.036S 

+ 

18. 3697 

-0.42235 

-0.00422 

— 

0. 52517 

- 0.87699 

-0.08063 

-0.151265 

+0.00777 


3. 87923 

+5.48 

+3.2298 

+ 

3. 82 

+ 6.8009 

-0. 0428 

-0. 7425 

+6.6 

+205. 2346 



+ 0. 4203 

- 3. 2365 

-0.1462 

+0.3070 

+0. 5112 

+ 

4. 8529 



+ 

0. 6773 

- 0.1608 

+0. 0271 

+0. 2400 

-1.1107 

— 

2. 3794 



+ 0.0343 

+ 0.2793 

+0.0795 

+0.0009 

-0.0315 

— 

0.8430 



+ 

0. 4078 

+ 2.4734 

+0. 2026 

-0.1287 

-2. 2924 

— 

5. 7295 

+0.1920 

+0.0019 

+ 

0. 2387 

+ 0.3986 

+0.0366 

+0.0688 

-0.0035 

+ 

1. 7631 

+5. 6720 

+3.2317 

+ 

5.5984 

+ 6.5549 

+0.1568 

-0. 2545 

+3. 6731 

+202.8988 

-0.036623 

-0.020867 

— 

0.036148 

- 0.042324 

-0.001012 

+0.001643 

-0.023717 

— 

1.310087 

+2 

+0.02 

+ 

2 

+ 0.67 

+0.02 

+0.39 

+2.8 

+ 38. 76 



+ 0.2435 

+ 1.4764 

+0.1209 

-0. 0768 

-1.3684 

— 

3. 4201 

-1 

-0. 01 

— 

1. 2435 

- 2.0764 

-0.1909 

-0. 3582 

+0.0184 

— 

9. 1849 

-0. 6490 

-0.3698 

— 

0. 6405 

- 0. 7500 

-0.0179 

+0.0291 

-0. 4203 

— 

23. 2149 

+0.3510 

-0. 3598 

+ 

0. 3595 

- 0.6800 

-0.0679 

-0.0159 

+ 1.0297 

+ 

2.9401 

-0.17795 

+0.18241 


0.18226 

+ 0.34474 

+0.03442 

+0.00806 

-0. 52203 


1. 49054 


91S65°—15 


-o 





























64 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Solution of 



17 

18 

29 

19 

20 

21 

22 


+6 

+2 

+ 

8. 60 

_2 




15 

-0. 8447 

-0. 8447 

— 

0. 00S4 





28 

-0.2077 

-0. 2077 

— 

0.1184 





16 

-0. 0625 

-0. 0625 

+ 

0. 0640 






+4.8851 

+0. 8851 

+ 

8. .5372 

-2 





Cn 

-0.18118 

— 

1. 74760 

+0.40941 






+6 

_ 

2. 30 

+2 

-2 




15 

-0. 8447 

— 

0. 0084 






28 

-0. 2077 

— 

0.1184 






16 

-0. 0625 

+ 

0. 0640 






17 

-0.1604 

— 

1. 5468 

+0.3624 






+4. 7247 

_ 

3. 9096 

+2.3624 

-2 





C 18 

+ 

0. 82748 

-0. 50001 

+0. 42331 






+127.4272 

-8.18 

+0.95 





15 

— 

0. 0001 







28 

— 

0.0674 







16 

— 

0. 0656 







17 

— 

14. 9196 

+3. 4952 






18 

— 

3. 2351 

+1. 9548 

-1.6550 






+109.1394 

-2.73 

-0. 7050 







C 29 

+0. 025014 

+0. 006460 








+6 

-2 







17 

-0. 8188 








18 

-1.1812 

+ 1 







29 

-0.0683 

-0. 0176 








+3. 9317 

-1. 0176 








C 19 

+0.25882 









+6 

+2 

-2 






18 

-0.8466 








29 

-0.0046 








19 

-0. 2634 









+4.8854 

+2 

-2 


~ 





C 20 

-0.40938 

+0.40938 








+6 

+2 







20 

-0.8188 

+0.8188 








+5.1812 

+2.8188 








C 21 

-0. 54404 









+6 








20 

-0. 8188 








21 

-1. 5335 









+3. 6477 









C 22 



























APPLICATION OF LEAST SQUARES TO TRIANGULATION 


65 


normals —Continued 


30 

31 

32 

33 • 

34 

V 

V 


+2 

+ 0.67 

+0.02 

+0. 39 

- 5.1 

+22. 06 


-1.0503 

- 1.7540 

-0.1613 

-0. 3025 

+ 0. 0155 

- 7. 7584 


-0. 2050 

- 0.2401 

-0. 0057 

+0. 0093 

- 0.1345 

- 7. 4308 


-0.0640 

+ 0.1210 

+0. 0121 

+0.0028 

- 0.1832 

- 0.5232 


+0. 6807 

- 1.2031 

-0.1349 

+0. 0996 

- 5.4022 

+ 6. 3475 


-0.13934 

+ 0. 24628 

+0. 02761 

-0. 02039 

+ 1.10585 

- 1.29936 

- 4.38 

+4 

- 3.00 

+0. 08 

+0.85 

- 1.7 

+ 11.03 


-1.0503 

- 1.7540 

-0.1613 

-0. 3025 

+ 0. 0155 

- 7. 7584 


-0. 2050 

- 0. 2401 

-0. 0057 

+0. 0093 

- 0. 1345 

- 7.4308 


-0.0640 

+ 0. 1210 

+0. 0121 

+0. 0028 

- 0.1832 

- 0.5232 


-0.1233 

+ 0.2180 

+0. 0244 

-0.0180 

+ 0.9788 

- 1.1500 

- 4.38 

+2. 5574 

— 4. 6551 

-0.0505 

+0. 5416 

- 1.0234 

- 5.8325 

+ 0. 92704 

-0.54128 

+ 0.98527 

+0. 01069 

-0.11463 

+ 0.21661 

+ 1.23447 

+ 8.0144 

-0.93 

-37. 6627 

-1.5212 

+0. 7206 

-18.4 

+79. 9881 


-0. 0105 

- 0.0175 

-0.0016 

-0. 0030 

+ 0. 0002 

- 0. 0776 


-0.1168 

- 0. 1368 

-0. 0033 

+0. 0053 

- 0.0766 

- 4. 2339 


+0. 0656 

- 0.1240 

-0. 0124 

-0. 0029 

+ 0. 1878 

+ 0. 5363 


—1.1896 

+ 2.1025 

+0. 2358 

-0. 1741 

+ 9. 4409 

-11.0929 

- 3.6244 

+2.1162 

- 3.S520 

-0.0418 

+0. 4482 

- 0.8468 

- 4.8263 

+ 4.3900 

-0. 0651 

-39. 6905 

-1.3445 

+0.9941 

- 9.6945 

+60. 2939 

- 0.040224 

+0.000596 

+ 0.363668 

+0. 012319 

-0.009109 

+ 0.088827 

- 0.552449 

- 4.38 

+2 

- 3.27 

+0.03 

+0. 23 

+ 2.1 

- 7.47 


+0.2787 

- 0.4926 

-0. 0552 

+0. 0408 

- 2.2117 

+ 2.5987 

+ 2.19 

-1.2787 

+ 2.3276 

+0. 0252 

-0. 270S 

+ 0.5117 

+ 2.9163 

+ 0.1098 

-0. 0016 

- 0.9928 

-0. 0336 

+0. 0249 

- 0.2425 

+ 1.5082 

- 2. 0S02 

+0. 9984 

- 2.4278 

-0. 0336 

+0. 0249 

+ 0.1575 

- 0.4467 

+ 0.52908 

-0. 25394 

+ 0.61749 

+0. 00855 

-0.00633 

- 0.04006 

+ 0. 11361 

+ 4.50 

-4 

+ 0. 19 

-0.35 

-0. 18 

+ 2.3 

+ 5.41 

- 1.8541 

+ 1.0826 

- 1.9706 

-0. 0214 

+0. 2293 

- 0.4332 

- 2.4690 

+ 0.0284 

-0.0004 

- 0.2564 

-0.0087 

+0. 0064 

- 0.0626 

+ 0.3895 

- 0.5384 

+0. 2584 

- 0.6284 

-0.0087 

+0.0064 

+ 0.0408 

- 0.1157 

+ 2.1359 

-2.6594 

- 2.6654 

-0.3S88 

+0. 0621 

+ 1.8450 

+ 3.2148 

- 0.43720 

+0.54436 

+ 0.54558 

+0. 07958 

-0.01271 

- 0.37766 

- 0.65804 

-10. 92 

+ 1 

- 0.93 

-0.21 

-0.01 

+ 6.1 

+ 5.01 

- 0.8744 

+1.0887 

+ 1.0912 

+0.1592 

-0.0254 

- 0.7553 

- 1.3161 

-11.7944 

+2.0887 

+ 0.1612 

-0. 0508 

-0. 0354 

+ 5.3447 

+ 3. 7140 

+ 2.27638 

-0. 40313 

- 0.03111 

+0. 00980 

+0. 006S3 

- 1.03156 

— 0.71682 

- 3. 75 

+3 

+ 3.12 

+0.11 

-0. 06 

+ 0.1 

+ 8.52 

+ 0.8744 

-1.0887 

- 1.0912 

-0.1592 

+0. 0254 

-f~ 0. / 553 

+ 1.3161 

+ 6.4166 

-1.1363 

- 0.0877 

+0. 0276 

+0. 0193 

- 2.9077 

- 2.0206 

+ 3.5410 

+0. 7750 

+ 1.9411 

-0.0216 

-0.0153 

- 2.0524 

+ 7.8155 

- 0.97075 

-0. 21246 

- 0.53214 

+0. 00592 

+0. 00419 

« 

+ 0. 56266 

- 2.14258 


» 
































66 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Solution of normals —Continued 



30 

31 

32 

33 

34 


2 


+76.0764 


9.74 

+ 

1.8338 


0.5171 


0.0430 


8.2 

+ 

48.4945 

18 

- 4.0604 

+ 

2.3708 

— 

4.3155 

— 

0.0468 

+ 

0.5021 

— 

0.9487 

— 

5.4070 

29 

- 0.1766 

+ 

0.0026 

+ 

1.5965 

+ 

0.0541 

— 

0.0400 

+ 0.3900 

— 

2.4253 

19 

- 1.1006 

+ 

0.5282 

— 

1.2845 

— 

0.0178 

!+ 

0.0132 

+ 

0.0833 

— 

0.2363 

20 

- 0.9338 

+ 

1.1627 

+ 

1.1653 

+ 

0.1700 

i— 

0.0272 

— 

0.8066 

— 

1.4055 

21 

-26.8485 

+ 

4.7547 

+ 

0.3670 

— 

0.1150 

— 

0.0806 

+ 12.1666 

+ 

8.4545 

22 

- 3.4374 


0. 7523 

— 

1.S843 

+ 

0.0210 

+ 

0.0149 

+ 

1.9924 

— 

7.5869 


+39.5191 


1.6733 

_ 

2.5217 

_ 

0.4522 

+ 

0.3394 

+ 

4.6770 

+ 

39.8883 


Cso 

+ 

0.042342 

+ 

0.003810 

+ 

0.011443 


0.008588 

— 

0.118348 

— 

1.009342 



+16 

+ 

6.82 

+ 

1.67 

+ 

3.69 

_ 

7.3 

+ 

46.82 


1 

— 

1.5 

+ 

3.33 

+ 

1.13 

— 

1.54 

+ 

2.75 

+ 

0.9050 


3 

— 

2.25 

— 

10.17 

— 

1.8262 

— 

1.1475 

+ 

5.4375 

— 

15.3450 


23 

— 

0.0548 

— 

0.4201 

— 

0.0875 

— 

0.0038 

+ 

0.3218 

— 

0.8488 


4 

— 

0.0409 

+ 

0.2968 

+ 

0.0652 

+ 

0.1186 

+ 

0.4145 

+ 

0.3478 


6 

— 

0.0072 

— 

0.2617 

— 

0.0451 

+ 

0.0045 

+ 

0.0038 

— 

0.4591 


24 

— 

0.0118 

— 

1.1268 

— 

0.1772 

+ 

0.0897 

— 

0.1335 

— 

2.6783 


7 

— 

0.0007 

— 

0.0595 

— 

0.0122 

— 

0.0101 

+ 

0.0474 

— 

0.1546 


5 

— 

0.0024 

— 

0.0248 

— 

0.0085 

— 

0.0202 

+ 

0.0670 

+ 

0.0333 


25 

— 


— 

0.0001 

— 


— 


— 

0.0098 

— 

0.0033 


8 

— 

0.0005 

— 

0.0381 

— 

0.0069 

— 

0.0014 

+ 

0.0387 

— 

0.C314 


9 

— 

0.8766 

— 

0.7930 

— 

0.5382 

— 

0.9400 

+ 

1.6974 

— 

7.6127 


26 

— 

0.0243 

+ 

0.3037 

+ 

0.0353 

— 

0.0213 

— 

0.2595 

+ 

0.1644 


10 

— 

0.2291 

+ 

0.8493 

+ 

0.0819 

— 

0.0478 

+ 

0.8323 

— 

0.1252 


27 

— 

0.3457 

+ 

2.6618 

+ 

0.1202 

— 

0.2525 


0.4204 

— 

3.9911 


11 

— 

1.6295 

+ 

0.3869 

— 

0.0653 

— 

0.5774 

+ 

2.6724 

+ 

5.7248 


13 

— 

0.0042 

— 

0.0338 

— 

0.0096 

— 

0.0001 

+ 

0.0038 

+ 

0.1021 


14 

— 

0.0720 

— 

0.4404 

— 

0.0361 

+ 

0.0229 

+ 

0.4082 

+ 

1.0202 


15 

— 

1.3060 

— 

2.1810 

— 

0.20C5 

— 

0.3762 

+ 

0.0193 

— 

9.6472 


28 

— 

0.2024 

— 

0.2369 

— 

0.0057 

+ 

0.0092 


0.1328 

— 

7.3344 


16 

— 

0.0655 

+ 

0.1239 

+ 

0.0124 

+ 

0.0029 

_ 

0.1S77 

— 

0.5,59 


17 

— 

0.C948 

+ 

C.1676 

+ 

0.0188 

— 

0.0139 

+ 

0.7527 

— 

0.8845 


18 

— 

1.3843 

+ 

2.5197 

+ 

0.C273 

— 

0.2932 

+ 

0.5539 

+ 

3.1570 


29 

— 


— 

0.0237 

— 

0.0008 

+ 

0.0006 

— 

0.0058 

+ 

0.0359 


19 

— 

0.2535 

+ 

0.0165 

+ 

0.0085 

— 

0.0063 

_ 

0.0400 

+ 

0.1134 


20 

— 

1.4477 

— 

1.4509 

— 

0.2116 

+ 

0.0338 

+ 

1.0043 

+ 

1.7500 


21 

— 

0.8420 

— 

0.0650 

+ 

0.0205 

+ 

0.0143 


2.1546 

— 

1.4972 


22 

— 

0.1647 

— 

C. 4124 

+ 

0.0046 

+ 

0.0033 

+ 

0.4361 

_ 

1.6605 


30 

— 

0.07C9 

— 

0.1068 

— 

0.0191 

+ 

0.0144 

+ 

0.1980 

+ 

1.6890 



+ 

3.1119 

+ 

0.2312 

— 

0.0558 

_ 

1.3082 

+ 

7.0750 

+ 

9.0541 




C 31 

— 

0.07430 

+ 

0.01793 

+ 

0.42039 


2.27353 


2.90951 





+351.4744 

+49.3161 


L3.0822 

_ 

5.2 

+478.9301 




1 

— 

7.3926 

— 

2.5086 

+ 

3. 4188 

— 

6.1050 

— 

2.0091 




2 

— 

0.1387 

— 

0.0253 

— 

0.4881 

+ 

0.9406 

+ 

1.9049 




3 

— 

45.9684 

— 

8.2546 

— 

5.1867 

+24.5775 


69.3594 




23 

_ 

3.2189 

— 

0.6708 

— 

0.4889 

+ 

2.4655 

_ 

6.5041 




4 

— 

1.8798 

— 

0.4131 

— 

0.7508 


2.6254 

— 

2.2030 




6 

— 

9.4868 

— 

1.0349 

+ 

0.1021 

+ 

2.3146 

_ 

16.6447 




24 

-107.9561 


L6.9750 

+ 

8.5969 

-12.7883 

-256.6033 




7 

— 

4.8313 

— 

0.9947 

— 

0.8189 

+ 

3.8530 

— 

12.5558 




5 

— 

0.2567 

— 

0.0878 

— 

0.2085 

+ 

0.6923 

+ 

0.3441 




25 

— 

0.0008 

— 

0.0002 

— 

0. C003 


0.0677 


0.0227 




8 

— 

2.7841 

— 

0.5031 

— 

0.1022 

+ 

2.8267 

— 

2.2937 




9 

— 

0.7173 

— 

0.4869 

— 

0.8509 

+ 

1.5355 

— 

6.8866 




26 

— 

3.8041 

— 

0.4424 

+ 

0.2674 

+ 

3.2497 

— 

2.0596 




10 

— 

3.1483 

— 

0.3035 

+ 

0.1770 


3.0853 

+ 

0.4639 




12 

— 

2.9963 

+ 

0.0353 

+ 

0.6148 

+ 

0.7067 


6.1198 




27 

— 

20.4950 

— 

0.9258 

+ 

1.9442 

+ 

3.2370 

+ 

30. 7307 




11 

— 

0.0919 

+ 

0.0155 

+ 

0.1371 


0.6346 


1.3594 




13 

— 

0.2751 

— 

0.0783 

— 

0.0009 

4- 

0.0310 

+ 

0.8303 




14 

— 

2.6710 

— 

0.2187 

+ 

0.1390 

+ 

2.4755 

+ 

6.1871 




15 

— 

3.6421 

— 

0.3348 

— 

0.6282 

+ 

0.0323 


16.1100 




28 

— 

0.2774 

— 

0.0066 

+ 

0.0108 

— 

0.1555 

_ 

8.5875 




16 

— 

0.2344 

— 

0.0234 

— 

0.0055 

+ 

0.3550 

+ 

1.0136 




17 

— 

0.2963 

— 

0.0332 

+ 

0.0245 


1.3305 

+ 

1.5633 




18 

— 

4.5805 

— 

0.0498 

+ 

0.5336 

_ 

1.0083 


5.7466 




29 

— 

14.4342 

— 

0.4890 

+ 

0.3615 

— 

3.5256 


21.9270 




19 

— 

1.4991 


0.0207 

+ 

0.0154 

+ 

0.0973 


0.2758 




20 

— 

1. 4542 

— 

0.2121 

+ 

0.0339 

+ 

1.0060 

+ 

1.7539 




21 

— 

0.0050 

+ 

0.0016 

+ 

0.0011 


0.1663 


0.1155 




22 

— 

1.0329 

+ 

0.0115 

+ 

0.0081 

+ 

1.0922 

_ 

4.1589 




30 

— 

0.1609 

— 

0.0289 

+ 

0.0217 

+ 

0.2984 

+ 

2.5453 




31 

— 

0.0172 

+ 

0.0041 

+ 

0.0972 

— 

0.5257 


0.6727 





+ 105.7210 

+ 13.0019 

— 

6.0470 

+ 14.5692 

+ 127.9051 






C 32 


0.129226 

+ 

0.057198 


0.137808 

— 

1.209830 

































APPLICATION OP LEAST SQUARES TO TRIANGULATION 


67 


Solution of normals —Continued 



33 

34 

>? 

y 


+ 8.7558 

- 1.0341 

- 2.5334 

+ 79.9111 

1 

- 0.8513 

+ 1.1601 

- 2.0717 

- 0.6818 

2 

- 0.0046 

- 0.0889 

+ 0.1714 

+ 0.3471 

3 

- 1.4823 

- 0.9314 

+ 4.4134 

- 12.4550 

23 

- 0.1398 

- 0.1019 

+ 0.5138 

- 1.3555 

4 

- 0.0908 

- 0.1650 

- 0.5770 

- 0.4841 

6 

- 0.2817 

+ 0.0279 

+ 0.3989 

- 2.8685 

24 

- 2.6691 

+ 1.3518 

- 2.0108 

- 40.3482 

7 

- 0.2048 

- 0.1686 

+ 0.7933 

- 2.5851 

5 

- 0.0300 

- 0.0713 

+ 0.2368 

+ 0.1177 

25 

— 

- 0.0001 

- 0.0143 

- 0. 0048 

8 

- 0.0909 

- 0.0185 

+ 0.5108 

- 0.4145 

9 

- 0.3305 

- 0.5776 

+ 1.0422 

- 4.6742 

26 

- 0.0514 

+ 0.0311 

+ 0.3779 

- 0.2395 

10 

- 0.0293 

+ 0.0171 

- 0.2974 

+ 0.0447 

12 

- 0.0004 

- 0.0072 

- 0.0083 

+ 0.0721 

27 

- 0.0416 

+ 0.0873 

+ 0.1454 

+ 1.3804 

11 

- 0.0026 

- 0.0231 

+ 0.1071 

+ 0.2294 

13 

- 0.0223 

- 0.0002 

+ 0.0088 

+ 0.2362 

14 

- 0.0179 

+ 0.0114 

+ 0.2027 

+ 0.5067 

15 

- 0.0308 

- 0.0578 

+ 0.0030 

- 1.4811 

28 

- 0.0002 

+ 0.0003 

- 0.0037 

- 0.2053 

16 

- 0.0023 

- 0.0005 

+ 0.0354 

+ 0.1012 

17 

- 0.0037 

+ 0.0027 

- 0.1492 

+ 0.1753 

IS 

- 0.0005 

+ 0.0058 

- 0.0109 

- 0.0623 

29 

- 0.0166 

+ 0.0122 

- 0.1194 

+ 0.7428 

19 

- 0.0003 

+ 0.0002 

+ 0.0013 

- 0.0038 

20 

- 0.0309 

+ 0.0049 

+ 0.1468 

+ 0.2558 

21 

- 0.0005 

- 0.0003 

+ 0.0524 

+ 0.0364 

22 

- 0.0001 

- 0.0001 

- 0.0122 

+ 0.0463 

30 

- 0.0052 

+ 0.0039 

+ 0.0535 

+ 0.4564 

31 

- 0.0011 

- 0.0261 

+ 0.1410 

+ 0.1804 

32 

- 1.7655 

+ 0.7814 

- 1.8827 

- 16.5287 


+ 0.5568 

+ 0.2254 

- 0.3351 

+ 0.4471 


C33 

- 0.40481 

+ 0 . 601 S 3 

- 0.80298 



+ 9 . 4562 

- 20.0499 

- 18.(3051 


1 

- 1.5811 

+ 2.8233 

+ 0.9291 


2 

- 1.7177 

+ 3.3105 

+ 6.7041 


3 

- 0.5825 

+ 2.7731 

- 7.8260 


23 

- 0.0743 

+ 0.3745 

- 0.9880 


4 

- 0.2999 

- 1.0486 

- 0.8799 


6 

- 0.0028 

- 0.0395 

+ 0.2844 


24 

- 0.0846 

+ 1.0184 

+ 20 . 4342 


7 

- 0.1388 

+ 0.6531 

- 2.1283 


5 

- 0.1694 

+ 0.5625 

+ 0.2796 


25 

-0. 0001 

- 0.0263 

- 0.0088 


8 

- 0.0038 . 

+ 0.1038 

- 0.0842 


9 

- 1.0094 

+ 1.8215 

- 8.1691 


26 

- 0.0188 

- 0.2284 

+ 0.1448 


10 

- 0.0100 

+ 0.1735 

- 0.0261 


12 

- 0.1262 

- 0.1450 

+ 1.2557 


27 

- 0.1844 

- 0.3071 

- 2.9152 


11 

- 0.2046 

+ 0.9469 

+ 2.0285 


13 

— 

+ 0.0001 

+ 0.0027 


14 

- 0.0072 

- 0.1288 

- 0 . 3220 


15 

- 0.1084 

+ 0.0056 

- 2.7787 


28 

- 0.0004 

+ 0.0060 

+ 0.3334 


16 

- 0.0001 

+ 0.0083 

+ 0.0237 


17 

-0.0020 

+ 0.1102 

- 0.1294 


18 

- 0.0621 

+ 0.1173 

+ 0 . 6(386 


29 

- 0.0091 

+ 0.0883 

- 0.5492 


19 

-0.0002 

- 0 . C 010 

+ 0.0028 


20 

- 0.0008 

- 0.0234 

- 0.0409 


21 

-0.0002 

+ 0.0365 

+ 0.0254 


22 

-0.0001 

- 0.0086 

+ 0.0327 


30 

- 0.0029 

- 0.0402 

- 0.3426 


31 

- 0.5500 

+ 2.9743 

+ 3.8063 


32 

- 0.3459 

+ 0.8333 

+ 7.3159 


33 

- 0.0912 

+ 0.1357 

- 0.1810 



+ 1.4672 

- 3.1701 

- 1.7029 



C34 

+ 2.16065 

+ 1.16065 





















68 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 25 


Bach solution 


34 

33 

32 

31 

30 

22 

21 

20 

+ 2.16065 

+ 0.60183 
0.87465 

- 0.13781 
+ 0.12358 
+ 0.03526 

- 2.27353 
+ 0 . 90 S 32 
- 0.00489 
- 0.00156 

- 0.11835 
- 0.01856 
- 0.00312 
+ 0.00134 
- 0 . 058 C 8 

+ 0.5627 
+ 0.0091 
- 0.0016 
-0.0112 
+ 0.2914 
+ 0.1910 

- 1.0316 
+ 0.0148 
- 0.0027 
- 0.0007 
+ 0 . 5 o 30 
- 0.4479 
— 0.5666 

- 0.3777 
- 0.0275 
- 0.0217 
+ 0.0115 
- 0.7467 
+ 0.0860 
+ 0.4263 
+ 0.6065 

+ 2.16065 

- 0.27282 

+ 0.02103 

- 1.37166 

- 0 . 19 C 77 

+ 1.0414 

- 1.4817 

- 0.0433 

19 

29 

18 

17 

16 

28 

15 

14 

- 0.0401 
- 0.0137 
- 0.0023 
+ 0.0130 
+ 0.3483 
- 0.1041 
-0.0112 

+ 0.08883 
- 0.01968 
- 0.00336 
+ 0.00765 
- 0.00082 
+ 0.00791 
- 0.00028 
+ 0.00475 

+ 0.2166 
- 0.2477 
- 0.0029 
+ 0.0207 
+ 0.7425 
- 0.1824 
- 0.0183 
- 0.0950 
+ 0.0703 

+ 1.1058 
- 0.0441 
- 0.0075 
+ 0.0052 
+ 0.1911 
+ 0.0777 
- 0 . 14 S 5 
- 0.0913 

- 0.5220 
+ 0.0174 
- 0.0094 
+ 0.0072 
+ 0.2500 
+ 0.0155 
- 0.0897 
- 0.1937 

- 0.02372 
+ 0.00355 
+ 0.00028 
- 0.00089 
+ 0.04958 
- 0.00177 
- 0.01845 
- 0.03986 
+ 0.06003 

+ 0.0078 
- 0.3268 
+0.0220 
- 0.0184 
+ 0.7204 
- 0.0004 
- 0.2128 
- 0.4597 
+ 0.2624 
+ 0.0028 

- 0.6842 
- 0.0830 
- 0.0165 
+ 0.0155 
- 0.1670 
+ 0.2141 
- 0.0197 
- 0.0005 

+ 0.1899 

+ 0.08500 

+ 1 . 08 S 4 

- 0.5247 

- 0.7413 

+ 0.5038 

+ 0.02875 

- 0.0027 

13 

11 

27 

12 

10 

26 

9 

8 

- 0.0342 
+0.0021 
- 0.0235 
+ 0.0064 
- 0.0511 
- 0.0088 
- 0.0005 
+ 0.1381 

- 1.1552 
+ 0.5393 
- 0.0077 
- 0.0035 
- 0.9662 
- 0.0084 
- 0.0014 
+ 0.3937 
- 0.0240 

- 0.06283 
- 0.08154 
- 0.00488 
+ 0 . 0 CS 37 
+ 0 . 0 / C 87 
+ 0.00181 
+ 0.00006 
- 0.01706 
- 0.00025 
- 0.10590 

+ 0.1667 
+ 0.3133 
- 0.0023 
- 0.0149 
+ 0.0095 
+ 0.4111 
+ 0.0108 

+ 0.8869 
-0.1100 
- 0.0238 
+ 0.0190 
+ 0.3349 
- 0.3013 
+ 0 . 1 S 40 

+ 0.69009 
+0.122 C8 
+ 0.02563 
- 0.01699 
- 0 . C 8847 
+ 0.07899 
- 0.10586 
+ 0.16213 

+ 0.8316 
- 0.9957 
+ 0.0719 
— 0.0682 
+ 0.5890 
- 0.5190 
+ 0.3402 
- 0.4948 
— 0.1404 

+ 0.7631 
- 0.0596 
+ 0.0371 
- 0.0158 
+ 0.0141 
+ 0.4017 
+ 0.2825 
+ 0.0613 

+ 0.8942 

+ 0.9897 

+ 0.0285 

+ 0.86820 

+ 1.4844 

- 1.2334 

- 0.3254 

- 0.19135 

25 

5 

7 

24 

6 

4 

23 

3 

+ 1.12809 
+ 0.01057 
- 0.00073 
+ 0.00026 
- 0.00251 
+ 0.18351 
- 0.00279 
+ 0.01272 

+ 0.7073 
- 0.4604 
+ 0.0245 
- 0.0055 
+ 0.0348 
- 0.0977 
- 0.1744 
+ 0 . 7955 
+ 0.3660 

+ 0.7675 
- 0.3524 
+ 0.0541 
-0.0202 
+ 0.0163 
-0. 6200 
— 0.5552 

- 0.10618 
+ 0 . 15423 ^ 
+ 0.03845 
- 0.01885 
+ 0.01283 
- 0.02483 
- 0.02413 
+ 0.00720 

+ 0.3296 
+ 0.0499 
+ 0.0635 
- 0.0284 
+ 0.0511 
+ 0.5654 
- 0.2745 
- 0.2731 
- 0.0264 

+ 0.8568 
+ 0.5294 
- 0.0368 
+ 0.0129 
+ 0.1328 
- 0.1783 
+ 0.4765 
- 0.0271 
+ 0.1830 

+ 1.15411 
- 0.49453 
+ 0.08567 
- 0.03169 
+ 0.26972 
- 0.03462 
- 0.11342 

- 1.8125 
+ 0.8264 
- 0.1661 
+ 0.0713 
- 1.0287 
+ 0.0766 
+ 0.9746 
+ 0.2662 

- 0.7099 

+ 0.83524 

+ 1.32912 

+ 0.03872 

- 0.7922 

+ 1.1901 

+ 0.4571 

+ 1.9492 

2 

1 

Probable error of an observed direction = ± 0 . 6745 - y /^^= ± 1 ". 2 

- 1.0938 
+ 1.2262 
- 0.0080 
+ 0.0034 
- 0.3169 
+ 0.3961 

+ 0.9167 
- 1.1091 
- 0.1028 
+ 0.0233 
+ 0.6858 
- 0.0738 
- 0.2641 
- 0.0690 

+ 0.2070 

[ 

+ 0.0070 



































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


69 


Computation of corrections 


- 0.007 
- 0.207 
- 0 . 459 
+ 1.372 


+ 0 . G 99 
+ 0.7 


+ 0 . 792 
+ 1.949 
+ 0.008 
+ 0.101 
- 1.372 
+ 0.094 
- 0.188 
- 0.886 


+ 0.498 
+ 0.5 


17 


+ 0.007 
+ 0.792 
+ 0.393 
- 1.372 
- 0.003 
+ 0.038 
+ 2.593 


+ 2.448 

+ 2.4 


10 


- 0.007 
- 0.792 
+0.200 
+ 1.372 
+ 0.049 
- 0.183 
- 0.756 


- 0.117 

- 0.1 


18 


+ 0.207 
- 0.792 
+ 0.067 
+ 0.003 
- 0.038 
- 2.593 


- 3 . 

- 3 . 


146 

2 


11 


+ 0.007 
- 0.209 
- 0.049 
+ 0.183 
+ 0.756 


+0.688 

+ 0.7 


19 


- 0.007 
+ 0.019 
- 0.106 
- 1.275 


- 1.369 

- 1.4 


12 


- 0.207 
+ 0.792 
- 0.359 
+ 0.048 
- 0.180 
+ 3.003 


+ 3.097 
+ 3.1 


20 


- 0.207 


- 0.207 
- 0.2 


13 


+ 0.207 
+ 0.952 


+ 1.159 

+ 1.2 


+ 0.007 
+ 0 . 207 
- 0.019 
+ 0.106 
+ 1.275 


+ 1.576 

+ 1.6 


14 


21 


- 0.792 
- 1.949 
- 0.593 
- 0.205 
+ 1.372 
- 0.160 
+ 0.371 
- 0 . 735 


- 0.174 
- 0.094 
+ 0.188 
+ 0 . 886 


+ 0.806 
+ 0.7 


15 


+ 0.317 

+ 0.172 

- 0.341 

- 1.620 


- 1.472 

- 1.4 


- 2.691 ! 
- 2.7 


22 


23 


- 1.949 

+ 0.073 


— 1 . S 76 

- 1.9 


16 


+ 1.949 
- 1.190 
- 0 . 457 
- 0.112 
- 0.359 
- 0.061 
+ 0.150 
- 0.648 


- 0.728 
- 0.7 


24 


+ 1.190 
- 1.484 
+ 0.325 
+ 0 . 984 
- 0.260 


+ 0.755 

+0.8 


+ 0.457 
- 0.625 


- 0.168 
- 0.1 


+ 1.484 
- 0.990 
+ 0.451 


- 0.894 
+ 0.804 


+ 0.945 

+ 1.0 


- 0.090 

- 0.1 


- 0.325 
+ 0.990 
+ 1.233 
- 0.191 
- 1.617 
- 1.372 
- 0.089 
+ 0.025 
+ 1.448 


- 1.233 
+ 0.894 
+ 0.813 
+ 0.089 
- 0.025 
- 1.448 


- 1.190 
+ 0.710 
- 0.532 
- 0.052 
+ 0.134 
- 0.735 


- 0.910 

- 0.9 


- 1.665 

- 1.6 


+ 0.102 
+ 0.1 


- 0.457 
- 0.710 
+ 1.781 


+ 0.614 
+ 0.7 


25 


26 


27 


28 


29 


30 


31 


32 


- 1.949 
+ 1.190 
+ 0.457 
- 0.029 
- 1.249 
+ 0.113 
- 0.254 
+ 0.151 


- 1.570 

- 1.5 


33 


- 0.457 


- 0.457 
- 0.4 


+ 1.949 
+ 0.141 


+ 2.090 

+2.1 


- 0.112 
- 0.061 
+ 0.120 
+ 0.583 


+ 0.530 

+ 0.5 


+ 0.325 
- 1.042 
- 0.089 
+ 0.207 
- 0.367 


- 1.484 
+ 1.667 


+ 0.710 

- 1.874 


+ 0.183 

+0.2 


- 1.164 

- 1.2 


- 0.966 
- 1.0 


34 


35 


36 


37 


38 


+ 0.457 
+ 0.710 


- 0.710 


+ 1.167 

+ 1.2 


- 0.710 

- 0.7 


- 1.484 
+ 0.990 


+ 1.484 


- 0.990 


- 0.494 
- 0.5 


+ 1.484 
+ 1.5 


- 0.990 
- 1.0 


- 1.190 
+ 1.484 
- 0.325 
+ 1.661 
- 0 . 625 
+ 0.067 
- 0.134 
- 0.627 

+ 1.190 
- 0.710 
+ 0.213 
+ 0.023 
- 0.074 
+ 0.994 

+ 1.636 
+ 1.6 

+ 0.311 

+ 0.3 

39 

40 

+ 0.003 
+ 0.017 
- 0.048 
+ 0.065 
- 0.346 

+ 1.233 
- 0.028 
+ 0 . 741 
- 0.003 
+ 0.031 
- 1.372 
+ 0.048 
- 0.055 
+ 0.346 

- 0.319 

- 0.3 


+ 0 . 941 
+ 0.9 










































































70 COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 28 


Computation of corrections —Continued 


41 

42 

43 

44 

45 

46 

47 

48 

- 0.741 
- 0.048 

+ 0.325 
- 0.990 
- 1.233 
- 0.148 
+ 1.372 
+0.012 
+ 0.005 
- 1.340 

+ 0.028 

+ 0.990 

+ 0.564 

- 0.325 
- 0.417 
-0.012 
- 0.005 
+ 1.340 

- 0.028 
+ 0.725 

- 0.894 
+ 0.028 
0.612 

+ 0.894 
- 0.113 

+ 0.028 
+0.0 

- 0 . 789 
-0.8 

+ 1.554 
+ 1.6 

+ 0.697 
+ 0.7 

+ 0.781 
+0.8 

- 1.478 

- 1.5 




+ 0.581 
+0.6 

♦ 



- 1.997 

-2.0 







49 

50 

51 

52 

53 

54 

55 

56 

+ 1.233 
- 0.894 
+0.002 
- 0.038 
+ 0.432 

+ 0.741 
+ 0.525 
-0.121 

+ 0.894 
- 0.028 
- 0.572 

- 1.233 
+ 0.028 
- 0.741 
+ 0.003 
+ 0.570 
+ 0.078 
+ 1.372 
+ 0.038 
- 0.432 

- 1.088 
- 0.434 

- 0.504 
+ 0.541 
- 0.051 
+ 0.027 
- 0.151 

- 0.003 
- 0.525 
+ 1.088 
+ 0.504 
+ 0.043 
- 0.107 
- 1.372 
+ 0.051 
- 0 . 027 
+ 0.151 

+ 0.741 

+ 0.741 
+ 0.7 

- 1.522 

- 1.5 

+ 1.145 
+ 1.2 

+ 0.294 
+ 0.3 

+ 0 . 735 
+0.8 


- 0.138 
-0.1 





- 0.317 

- 0.3 









- 0.197 
-0.2 


57 

58 

59 

60 

61 

62 

63 

64 

+ 0.525 

- 0.741 
- 0.525 

+ 0.003 
+ 0.525 
- 1.088 
- 0.504 
- 0.115 
- 0.109 
+ 1.372 
+ 0.037 
-0.022 
- 0.691 

- 0.525 
+ 0.263 

- 0.003 
- 0.149 
- 0.037 
+0.022 
+ 0.691 

+ 1.088 
- 0.190 
+ 0.295 

+ 0.504 
+ 0.190 
+ 0.043 
- 0.186 
+ 0.771 
- 1.372 
- 0 . 009 
- 0.027 
+ 0.151 

+ 1.482 
- 1.118 

+ 0.525 

+ 0.5 

- 1.266 

- 1.3 

- 0.262 
-0.2 

+ 0.364 
+ 0.4 

+ 1.193 
+ 1.2 




+ 0.524 

+0.6 







+ 0.065 
+0.1 




- 0.592 

-0.6 





65 

66 

67 

68 

69 

70 

71 

72 

- 0.043 
- 1.482 
+ 0.346 
+ 0.009 
+ 0.027 
- 0.151 

- 0.190 

- 1.088 
+ 0.190 

+ 1.088 

- 1.041 
+ 0.317 
- 0.024 

+ 0.043 
+ 1.041 
- 0.226 
- 1.372 
+ 0.024 

- 0.504 
- 0.190 
- 0 . 043 
- 0.105 
- 0.091 
+ 1.372 
+ 0.060 
- 0.019 
- 0.346 

+ 0.504 
+ 0.425 
-0. 060 
+ 0.019 
+ 0.346 

- 0.190 
-0.2 

+ 1.088 
+ 1.1 

— 0.898 

- 0.9 

- 0 . 748 
-0.8 




- 0 . 490 
- 0.5 

+ 1.234 
+ 1.2 

- 1.294 

- 1.3 











+ 0.134 
+0.1 


73 

74 

75 

76 

77 

78 

79 


+ 0.190 
- 0.320 

+ 0.043 
+ 1.482 
- 0.030 
+ 0.025 
- 0.030 
- 0.130 

- 0.043 
- 1.041 
-0. 626 
+ 1.372 
- 0.025 
+ 0 . 030 
+ 0.130 

- 1.482 
+ 1.041 
+ 0 . 655 
- 1.372 

+ 1.482 
- 1.041 
+ 0 . 004 

- 1.482 

+ 1.041 
- 0.004 


- 1.482 

- 1.5 


- 0.130 
-0.2 

+ 1.037 
+1.0 


+ 0.445 

+ 0.4 


- 1.158 

-1.2 




+ 1.360 
+ 1.3 






- 0 . 203 
-0.2 





















































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


71 


Final solution of triangles 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Logarithm 



O 

/ 

// 

// 

n 

n 

o / n 



Turn-Dundas 








4.266771 

-10+11 

Tower 

42 

00 

30.0 

+ 0.8 

30.8 

0.1 

30.7 

0.174417 

-1+2 

Turn 

24 

17 

25.4 

+ 1.7 

27.1 

0.2 

26.9 

9.614231 

-4+6 

Dundas 

113 

41 

59.6 

+ 3.0 

62.6 

0.2 

' 42 02. 4 

9.961733 






+ 5.5 


0.5 




Tower-Pundas 








4.055419 


Tower-Turn 








4.402921 


Turn-Dundas 








4.266771 

-12+13 

Lazaro 

25 

52 

38.1 

- 1.9 

36.2 

0.4 

35.8 

0.360081 

-1+3 

Turn 

110 

36 

08.7 

- 3.9 

04.8 

0.5 

04.3 

9.971300 

- 5+.6 

Dundas 

43 

31 

18.5 

+ 1.8 

20.3 

0.4 

19.9 

9.837989 






- 4.0 


1.3 




Lazaro-Dundas 








4. 598152 


Lazaro-Turn 








4.464841 


Turn-Tower 








4. 402921 

-12+14 

Lazaro 

42 

30 

21.5 

- 5.8 

15.7 

0.6 

15.1 

0.170282 

-2+3 

Turn 

86 

18 

43.3 

- 5.6 

37.7 

0.7 

37.0 

9. 999099 

- 9+10 

Tower 

51 

11 

09.1 

- 0.6 

08.5 

0.6 

07.9 

9.891638 






-12.0 


1.9 




Lazaro-Tower 








4. 572302 


Lazaro-Turn 








4. 464841 


Dundas-Tower 








4.05.5419 

-13+14 

Lazaro 

16 

37 

43.4 

- 3.9 

39.5 

0.4 

39.1 

0. 543408 

-4+6 

Dundas 

70 

10 

41.1 

+ L2 

42.3 

0.4 

41.9 

9.973475 

- 9+11 

Tower 

93 

11 

39.1 

+ 0.2 

39.3 

0.3 

39.0 

9.999325 

1 





- 2.5 


1.1 




Lazaro-Tower 








4. 572302 


Lazaro-Dundas 








4. 598152 


Lazaro-Tower 








4. 572302 


Tow Hill 



38.8 


37.7 

2.2 

21 40 35.5 

0. 432543 

-14+15 

Lazaro 

47 

10 

07.2 

+ 1.3 

08.5 

2.2 

06.3 

9.865314 

-7+9 

Tower 

111 

09 

20.5 

- 0.2 

20.3 

2.1 

18.2 

9.969699 








6.5 




Tow Hill-Tower 






4 


4.870159 


Tow Hill-Lazaro 








4.974544 


Lazaro-Tower 








4.572302 

—25+26 

Nichols 

30 

04 

51.8 

+ 3.6 

55.4 

1.7 

53.7 

0.299961 

— 14+16 

Lazaro 

101 

38 

55.1 

+ 2.0 

57.1 

1.7 

55. 4 

9. 990962 

-8+9 

Tower 

48 

16 

10.2 

+ 2.4 

12.6 

1.7 

10.9 

9.872905 






+ 8.0 


5.1 




Nichols-Tower 








4. 863225 


Nichols-Lazaro 








4. 745168+! 


Lazaro-Tow Hill 








4.974.544 

-25+27 

Nichols 

89 

23 

18.6 

+ 2.0 

20.6 

3.6 

17.0 

0.000025 

— 15+16 

Lazaro 

54 

28 

47.9 

+ 0.7 

48.6 

3.6 

45.0 

9.910573 


Tow Hill 



64.3 


61.6 

3.6 

36 07 58.0 

9. 770601 








10.8 




Nichols-Tow Hill 








4. 885142 


Nichols-Lazaro 








4. 745170 -1 




































72 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final solution of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Logarithm 



O 

/ 

// 

// 

rr 

// 

Off/ 



Tower-Tow Hill 








4.870159 

-26+27 

Nichols 

59 

18 

26.8 

- 1.6 

25.2 

4.1 

21.1 

0. 065550 

— 7+ 8 

Tower 

62 

53 

10.3 

- 2.6 

07.7 

4.0 

03.7 

9. 949433 


Tow Hill 



35.1 


39.3 

4.1 

57 48 35.2 

9.927516 








12.2 




Nichols-Tow Hill 








4.885142 


Nichols-Tower 

i 







4.863225 


Lazar o-Nichols 








4. 745169 

-31+32 

Ken 

116 

59 

49.5 

+ 1.3 

50.8 

0.8 

50.0 

0.050108 

-16+17 

Lazaro 

22 

32 

57.3 

+ 1.5 

58.8 

0.7 

58.1 

9. 583744 

-23+25 

Nichols 

40 

27 

12.5 

+ 0.1 

12.6 

0.7 

11.9 

9/812130 






+ 2.9 


2.2 




Ken-Nichols 







4. 379021 


Ken-Lazaro 








4.607407 


Lazaro-Nichols 








4. 745169 

-33+34 

Seel 

128 

55 

09.3 

+ 1.6 

10.9 

0.4 

10.5 

0.109005 

-16+18 

Lazaro 

38 

29 

18.8 

+ 0.6 

19.4 

0.5 

18.9 

9. 794041 

-24+25 

Nichols 

12 

35 

33.3 

-2.2 

31.1 

0.5 

30.6 

9. 338465 






+ 0.0 


1.4 




Seal-Nichols 







4. 648215 


Seal-Lazaro 








4.192639 

-33+35 

Lazaro-Ken 








4.607407 

Seal 

154 

32 

21.8 

- 0.3 

21.5 

0.2 

21.3 

0.366640 

-17+18 

Lazaro 

15 

56 

21.5 

- 0. 9 

20.6 

0 . 1 

20.5 

9. 438723 

-30+31 

Ken 

9 

31 

16.8 

+ 1.5 

18.3 

0 . 1 

18.2 

9. 218592 






+ 0.3 


0.4 




Seal-Ken 






4. 412770+ 1 


Seal-Lazaro 








4.192639 

-34+35 

Nichols-Ken 




• 




4. 379021 

Seal 

25 

37 

12.5 

- 1.9 

10.6 

0.4 

10.2 

0. 364122 

-23+24 

Nichols 

27 

51 

39. 2 

+ 2.3 

41.5 

0. 4 

41. 1 

9.669628 

-30+32 

Ken 

126 

31 

06.3 

+ 2. 8 

09. 1 

0.4 

08.7 

9. 905072 


Seal-Ken 




+ 3.2 


1.2 









4 412771 


Seal-Nichols 








4.648215 

-36+37 

Lazaro-Ken 








4 607407 

Mid 

128 

20 

51.4 

+ 2.0 

53.4 

0.3 

53. 1 

0 105542 

-17+19 

Lazaro 

35 

17 

43.3 

+ 0. 2 

43. 5 

0. 3 

43. 2 

9.761771 

9. 449655 

-29+31 

Ken 

16 

21 

23.9 

+ 0 . 1 

24. 0 

0. 3 

23. 7 






+ 2.3 


0.9 




Mid-Ken 

Mid-Lazar o 






4.474720 

4.162604 



-42+44 

Lazaro-Mid 

Round 

51 

05 

11 . 2 

+ 3.6 

14.8 

0.2 

14. 6 

4.162604 

0 108962 

— 19—f— 21 

Lazaro 

43 

39 

34.4 

- 0. 9 

33. 5 

0 . 2 

33. 3 

9 839081 

+36-38 

Mid 

So 

15 

11. 7 

+ 0. 5 

12 . 2 

0 . 1 

12 . 1 

9. 99850S 


Round-Mid 

Round-Lazaro 




+ 3.2 


0.5 









4.110647 

4. 270074 







































APPLICATION OF LEAST SQUARES TO T El A N G U L A*TI 0 N 
Final solution of triangles —Continued 


73 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Logarithm 




O 

f n 

// 

n 

// 

o / // 



Lazar o-Ken 








4 607407 

-42+45 

Round 

74 

42 

34.2 

+ 2.6 

36.8 

0.6 

36. 2 

0. 015651 

—17+21 

Lazaro 

78 

57 

17. 7 

- 0.7 

17.0 

0. 7 

16. 3 

9. 991879 

—28+31 

Ken 

26 

20 

06.8 

+ 1.3 

08. 1 

0.6 

07.5 

9. 647016 






+ 3.2 


1. 9 




Round-Ken 








4 614937 


Round-Lazaro 








4. 270074 


Mid-Ken 








4 474720 

—44+45 

Round 

23 

37 

. 23. 0 

- 1.0 

22.0 

0. 2 

21.8 

6. 397167 

—3/ +38 

Mid 

146 

23 

56.9 

- 2.5 

54. 4 

0. 1 

54. 3 

9. 743050 

-28+29 

Ken 

9 

58 

42.9 

+ 1.2 

44.1 

0.2 

43.9 

9. 238760 






- 2.3 


0. 5 




Round-Ken 








4. 614937 


Round-Mid 








4.110647 


Lazaro-Round 








4. 270074 

-49+52 

Cat 

89 

47 

52.8 

- 1. 1 

51.7 

0.2 

51.5 

0. 000003 

-21+22 

Lazaro 

26 

22 

55.0 

- 1.0 

54.0 

0. 1 

53.9 

9. 647723 

-40+42 

Round 

63 

49 

17.6 

-2.9 

14.7 

0.1 

14.6 

9. 952995 






- 5.0 


0.4 




Cat-Round 








3.917800 


Cat-Lazaro 








4. 223072 


Round-Cat 








3. 917800 

-46+47 

Spur 

33 

20 

40.5 

-2.2 

38.3 

0.1 

38.2 

0. 259903 

-40+43 

Round 

111 

30 

09.4 

- 0. 9 

08.5 

0.0 

08.5 

9.968671 

-51+52 

Cat 

35 

09 

14.0 

- 0.6 

13.4 

0.1 

13.3 

9. 760250 


* 




- 3. 7 


0.2 




Spur-Cat 








4.146374 


Spur-Round 








3. 937953 


Round-Lazaro 








4. 270074 

-46+48 

Spur 

105 

41 

48.0 

+ 0. 1 

48. 1 

0. 1 

48.0 

0. 016506 

-42+43 

Round 

47 

40 

51. 8 

+ 2.0 

53. 8 

0. 1 

53. 7 

9. 868888 

-20+21 

Lazaro 

26 

37 

18.2 

+ 0.2 

18.4 

0.1 

18.3 

9. 651373 






+ 2.3 


0.3 




Spur-Lazaro 








4.155468 


Spur-Round 






4 


3. 937953 


Cat-Lazaro 








4.223072 

-47+48 

Spur 

72 

21 

07.5 

+ 2.3 

09.8 

0.1 

09.7 

0. 020934 

-49+51 

Cat 

54 

38 

38. 8 

- 0. 5 

38.3 

0 . 2 

38. 1 

9.911462 

-20+22 

Lazaro 

53 

00 

13. 2 

- 0. 8 

12. 4 

0.2 

12.2 

9. 902368 






+ 1.0 


0.5 




Spur-Lazaro 








4.155468 


Spur-Cat 








4.146374 


Cat-Round 








3. 917800 

-59+61 

Beaver 

49 

58 

12.0 

+ 1.2 

13.2 

0.1 

13.1 

0.115935 

-52+55 

Cat 

87 

34 

53.9 

+ 0. 1 

54.0 

0.0 

54.0 

9. 999613 

-39+40 

Round 

42 

26 

51.8 

+ 1.2 

53.0 

0 . 1 

52.9 

9. 829253 






+ 2.5 


0.2 




Beaver-Round 








4. 033348 


Beaver-Cat 








3. 862988 



















































74 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final solution of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Logarithm 



O 

/ 

// 

// 

// 

ft 

o r tr 



Round-Beaver 








4.033348 

—56+57 

Snipe 

52 

14 

18.1 

- 0.2 

17.9 

0.1 

17.8 

0.102063 

-39+41 

Round 

105 

37 

20.7 

- 0.5 

20.2 

0.0 

20.2 

9.983652 

-60+61 

Beaver 

22 

08 

21.3 

+ 0.8 

22.1 

0.1 

22.0 

9.576182 






+ 0.1 


, 0.2 




Snipe-Beaver 








4.119063 


Snipe-Round 








3.711593 


Round-Cat 




§ 




3.917800 

-56+58 

Snipe 

79 

10 

11.9 

- 2.0 

09.9 

0.0 

09.9 

0. 007806 

-40+41 

Round 

63 

10 

28.9 

- 1.7 

27.2 

0.0 

27.2 

9. 950551 

-50+52 

Cat 

37 

39 

24.5 

- 1.5 

23.0 

0.1 

22.9 

9. 785987 






- 5.2 


0.1 




Snipe-Cat 








3.876157 


Snipe-Round 








3.711593 


Beaver-Cat 








3.862988 

-57+58 

Snipe 

26 

55 

53.8 

- 1.8 

52.0 

0.0 

52.0 

0.343980 

-59+60 

Beaver 

27 

49 

50.7 

+ 0.4 

51.1 

0.0 

51.1 

9. 669189 

-50+55 

Cat 

125 

14 

18.4 

- 1.4 

17.0 

0.1 

16.9 

9. 912095 






- 2.8 


0.1 



. 

Snipe-Cat 








3.876157 


Snipe-Beaver 








4.119063 


Beaver-Cat 








3. 862988 

-67+68 

Khwain 

62 

13 

29.4 

+ 2. 0 

31.4 

0.1 

• 31.3 

0.053161 

-59+62 

Beaver 

58 

43 

17.2 

+ 1.8 

19.0 

0.0 

19.0 

9.931792 

-53+55 

Cat 

59 

03 

08.4 

+ 1.3 

09.7 

0.0 

09.7 

9. 933305 






+ 5.1 


0.1 




Khwain-Cat 








3. 847941 


Khwain-Beaver 








3.849454 


Beaver-Cat 








3.862988 

-71+72 

Lim 

36 

34 

55.5 

+ 1.1 

56.6 

0.0 

56.6 

0.224770 

-59+63 

Beaver 

102 

39 

45.5 

+ 0.7 

46.2 

0.0 

46.2 

9.989306 

-54+55 

Cat 

40 

45 

17.4 

- 0.1 

17.3 

0.1 

17.2 

9. 814795 






+ 1.7 


0.1 




Lim-Cat 








4.077064 


Lim-Beaver 








3.902553 


Beaver-Khwain 








3. 849454 

-71+73 

Lim 

59 

25 

24.7 

- 0.3 

24.4 

0.0 

24.4 

0. 065022 

-62+63 

Beaver 

43 

56 

28.3 

- 1.1 

27.2 

0.0 

27.2 

9.841307 

-66+67 

Khwain 

76 

38 

09.2 

- 0.7 

08.5 

0.1 

08.4 

9. 988077 






- 2.1 


0.1 




Lim-Khwain 








3. 755783 


Lim-Beaver 








3.902553 


Cat-Khwain 








3. 847941 

-72+73 

Lim 

22 

50 

29.2 

- 1.4 

27.8 

0.1 

27.7 

0. 410972 

-53+54 

Cat 

18 

17 

51.0 

+ 1.4 

52.4 

0.0 

52.4 

9. 496870 

-66+68 

Khwain 

138 

si 

38.6 

+ 1.3 

39.9 

0.0 

39.9 

9. 818151 






+ 1.3 


0.1 




Lim-Khwain 








3. 755783 


Lim-Cat 








4.077064 
































APPLICATION OP LEAST SQUARES TO TRIANGULATION 


75 


Final solution of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Logarithm 



0 

f 

n 

// 

tf 

// 

Of ff 



Beaver-Lim 








3.902553 

-74+75 

South Twin 

60 

40 

06.2 

- 1.5 

04.7 

0.0 

04.7 

0. 059585 

-63+65 

Beaver 

78 

24 

38.4 

- 1.4 

37.0 

0.1 

36.9 

9.991054 

-70+71 

Lim 

40 

55 

17.8 

+ 0.6 

18.4 

0.0 

18.4 

9. 816260 






- 2.3 


0.1 




South Twin-Lim 








3. 953192 


South Twin-Beaver 








3. 778398 


South Twin-Beaver 








3. 778398 

-77+78 

Ham 

35 

38 

30.7 

- 1.9 

28.8 

0.0 

28.8 

0. 234548 

-74+76 

South Twin 

94 

10 

29.2 

- 2.5 

26.7 

0.1 

26.6 

9. 998847 

-64+65 

Beaver 

50 

11 

06.3 

- 1.7 

04.6 

0.0 

04.6 

9. 885424 


Ham-Beaver 
Ham-South Twin 




- 6.1 


0.1 

{ 

4.011793 

3. 898370+1 

3. 898371 


South Twin-Lim 








3. 953192 

-77+79 

Ham 

85 

04 

23. 4 

+ 0.6 

24.0 

0.1 

23.9 

0 . 001608 

-75+76 

South Twin 

33 

30 

23.0 

- 1.0 

22.0 

0.0 

22.0 

9. 741959 

-69+70 

Lim 

61 

25 

13.8 

+ 0.3 

14.1 

0.0 

14.1 

9. 943571 






- 0.1 


0.1 




Ham-Lim 








3. 696759 


Ham-South Twin 








3. 898371 


Beaver-Lim 








3. 902553 

-78+79 

Ham 

49 

25 

52.7 

+ 2.5 

55.2 

0.0 

55.2 

0.119395 

-63+64 

Beaver 

28 

13 

32.1 

+ 0.3 

32.4 

0.0 

32.4 

9. 674811 

-69+71 

Lim 

102 

20 

31.6 

+ 0.9 

32.5 

0.1 

32.4 

9. 989845 






+ 3.7 


0.1 




Ham-Lim 








3. 696759 


Ham-Beaver 








4. 011793 

























76 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final 'position computation , 

STATION TOWER 


Second angle 

a 

Joe 


J(f> 

F 


s 

COS os. 

B 


1st term 
2d term 


3d and 4th 
terms 

— J<t> 


Turn to Dundas 
Dundas and Tower 

Turn to Tower 


Tower to Turn 


54 


48 

12 


06. 742 
39. 415 


54 


4. 402921 
9.9676448 
8. 5097251 


2. 8802909 

It 

+ 759. 0859 
+ 0.3175 


+759. 4034 
+ 0.0115 


+ 759. 4149 

o tit 

54 41 47 


35 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec 4>' 


JX 


27.327 
-1 


First angle of triangle 
Turn 


8. 80585 
9.14128 
1. 55459 


Tower 


(<Jg) 2 


9. 50172 

n 

+0.0134 
-0. 0019 


-6 

4. 402921 
9. 5706188 
8. 5087480 
0. 2370138 


2. 7193010 

It 

+523.9635 


Arg. 

s 

JX 


Corr. 


X 

JX 

X' 


5. 7609 
2.3672 


8.1281 


-11 
+ 5 


- 6 


o 

357 
+ 24 


/ 

33 

17 

II 

07.7 

27.1 

21 


50 

34.8 

— 


7 

07.6 

180 




201 


43 

27.2 

42 


00 

30.8 

130 


56 

04. 052 

+ 


8 

43. 963 

131 

04 

48.015 

-h 


2 

.8803 

s 2 sin 2 

CL 

7. 9471 

E 


6. 4574 



7. 2848 

JX 


2. 7193010 

sin 

9.9117440 

sec h(J<i>) 


7 

—Ja 

2.6310457 

ft 

+427. Cl 


STATION LAZARO 











0 


1 

It 

a. 

Turn to Tower 






21 

50 

31.8 

Second angle 

Tower and Lazaro 





+ 86 


18 

37. 7 

a 

Turn to Lazaro 






108 

09 

12.5 

Jot 









— 

21 

10.8 










180 




a' 

Lazaro to Turn 






287 

48 

01.7 







First angle of triangle 

42 

30 

15. 7 


O 


/ 


It 









54 


48 

06. 742 


Turn 

X 

130 

56 

04. 052 

J<f> 

+ 


4 

51.079 



J\ 

+ 

25 

54. 366 

F 

54 


52 

57.821 


Lazaro 

X' 

131 

21 

58. 418 






-1 







-1 

s 

4. 464841 

s 2 

8.92963 



-h 


2. 4681 

COS a 

9. 4935464 

sin 2 a 

9.95564 

(3gY 2 

4.9281 

s 2 sin 2 

CX 

8. 8853 

B 

8. 5097251 


C 

1. 55459 

2. 3672 

E 


6. 4574 

h 

2. 4681125 



0. 43986 


7. 2953 



7. 8108 

1st term 

-293. 8411 

3d term 

+0.0020 







2d term 

+ 2.7534 

4th term 

+0. 0065 








-291.0877 











3d and 4th 1 
terms / 

+ 0.0085 


s 

+ 27 
4. 464841 










G Q77C0£7 

Arg. 


JX 


3.1915533 










— J4> 

-291.0792 


V' 

8. 5087409 

$ 

-15 

sin 

9.9125251 

O / 

It 

sec F 

0. 2401420 

JX 

+42 

sec i(J<fi) 



htt+F) 

54 50 32. 3 



3.1915533 

Corr. 

+27 



3.1040784 




J\ 

+ 1554. 3661 



—Ja 


+ 1270.8 


































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


primary triangulation 


STATION TOWER 


o 

t 

tr 

177 

33 

43.6 

-113 

42 

02.6 

63 

51 

41.0 

— 

7 

43.0 

180 



243 

43 

58.0 

130 

55 

20. 042 

+ 

9 

27. 973 

131 

04 

48.015 


Third angle 

a 

Act 


* 

j(f> 




Dundas to Turn 
Tower and Turn 

Dundas to Tower 


Tower to Dundas 


54 


54 


38 

2 


35 


09.559 
42. 233 


27. 326 


Dundas 

Tower 


X 

JA 

y 


s 

4.055419 

s 2 

8.11087 



-h 

2. 2091 

COS a 

9. 6439895 

sin 2 a 

9.90630 

(36) 2 

4. 4203 

s 2 sin 2 a 

8 . 0172 

B 

8 . 5097372 

C 

1. 55194 

D 

2. 3681 

E 

6 . 4528 

h 

2. 2091457 


9. 56911 


6 . 7884 


6 . 6791 

1 st term 

+ 161.8623 

3d term 

+0. 0006 





2 d term 

+ 0.3708 

4 th term 

-0.0005 






+162. 2331 







3d and 4 th \ 

+0.0001 

s 

+3 
4.055419 





VvlIUo J 


sin a 

9. 9531462 

Arg. 


AX 

2. 7543283 

— A4> 

+ 162.2332 

A' 

8 . 5087480 

s 

-2 

sin h(4>+4>') 

9. 9112981 


o t n 

sec 4V 

0. 237013S 

AX 

+5 

sec %(A<t>) 



54 36 48. 4 


2. 7543273 

Corr. 

+3 


2. 6656264 


// 


ff 




// 



AX 

+567.9725 



—Aa 

+ 463.05 


STATION LAZARO 


O 

201 
- 51 

t 

43 

11 

// 

27.2 

08.5 

150 

32 

18. 7 

— 

14 

01.3 

180 



330 

18 

17.4 

131 

04 

48. 015 

+ 

17 

10. 401 

, 131 

21 

58. 416 
+ 1 

-h 3.0219 


Third angle 

a 

Act 


J(j) 


4>' 


s 

cos « 
B 


1 st terra 
2 d term 


3d and 4th \ 
terms / 

*(*+*') 


Tower to Turn 
Lazaro and Turn 

Tower to Lazaro 


Lazaro to Tower 


54 


+ 


35 

17 


54 


4. 572302 
9.9398618 
8 . 5097404 


52 


27. 326 
30. 494 


57. 820 


3.0219042 

n 

-1051.7298 
+ 1.2004 


-1050. 5294 
+ 0.0358 


-1050. 4936 

o / n 

54 44 12.6 


S 2 

sin 2 « 
C 


9.14459 
9. 38353 
1. 55122 


0. 07934 


3d term +0.0258 
4 th term +0.0100 


8 

sin a 
A' 

sec 4>' 


A X 


Tower 

Lazaro 

(<?g ) 2 


4. 572302 
9. 6918222 
8.5087409 
0. 2401420 


3. 0130064 

n 

+1030. 4012 


Arg. 

s 

AX 


Corr. 


X 

AX 


6.0428 
2. 3683 


8.4111 


-25 
+ 18 


- 7 


s 2 sin 2 a 
E 


AX 

sin §(<£+$') 
sec i(A<f>) 


-Act 


8. 5281 
6. 4516 


8. 0016 


3.0130064 
9. 9119609 


2. 9249673 
// 

+ 841.3 







































































COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation , 


STATION TOW HILL 











o 


/ 

n 

a 

Lazaro to Tower 





330 


18 

17.4 

Second angle 

Tower and Tow Hill 





+ 47 


10 

08.5 

CL 

Lazaro to Tow Hill 





17 

28 

25.9 

Ja 









— 

21 

07.5 










180 




a' 

Tow Hill to Lazaro 





197 

07 

18.4 







First angle of triangle 

21 

40 

37.7 


O 


I 


it 








<f> 

54 


52 

57. 820 


Lazaro 

X 

131 

21 

58.417 

J<j> 

— 


48 

32.022 



J\ 

+ 

25 

57. 250 

<y 

54 


04 

25. 798 

Tow Hill 

X' 

131 

47 

55. 667 
-2 

s 

4.974544 


s 2 

9.94909 



-h 


3. 4637 

COS a 

9.9794818 

sin 2 a 

8.95504 

(W 

6.9283 

s 2 sin 2 a 

8.9043 

B 

8 . 5097191 


C 

1. 55589 

D 

2.3667 

E 


6 . 4597 

h 

3. 4637449 



0. 46002 


9.2950 



8.8277 

1 st term 

+2909. 0080 

3d term 

+0.1972 







2 d term 

+ 2.8842 

4th term 

-0.0673 








+2911.8922 











3d and 4 th 1 

+ 0.1299 


s 

— 117 
4.974544 







terms / 



sin a. 

9. 4775129 

Arg. 


J\ 


3.1923584 

— J<j> 

+2912.0221 


A/ 

8 . 5087606 

s 

—159 

sin §(<£+<£') 

9.9105687 


O t 

tt 

sec 

0. 2315526 

J\ 

+ 42 

sec %(J<}>) 


108 


54 28 41. 9 



3.1923584 

Corr. 

-117 



3.1029379 







tr 



* 



It 




J\ 

+ 1557. 2502 



—Ja 

+ 1267.5 


STATION NICHOLS 








O 

/ 

// 

a 

Lazaro to Tow Hill 



17 

28 

25.9 

Second angle 

Tow Hill and Nichols 



+ 54 

28 

48.6 

a 

Lazaro to Nichols 



71 

57 

14.5 

Jet 






— 

40 

14.3 







180 



a' 

Nichols to Lazaro 



251 

17 

00.2 





First angle of triangle 

89 

23 

20.6 


O 

I 

n 






$ 

54 

52 

57.820 

Lazaro 

X 

131 

21 

58. 417 

A<j> 

— 

9 

27.129 


J\ 

+ 

49 

14. 369 

4’ 

54 

43 

30.691 

Nichols 

X' 

132 

11 

12. 786 


+1 


S 

COS a 

B 

4.745169 

9.4910534 
8.5097191 

$ 2 

sin 2 a 

C 

9.49034 

9.95619 

1. 55589 

(5g) 2 

5.5072 
2.3667 

-h 

s 2 sin 2 a 

E 

h 

2.7459415 


1.00242 


7.8739 



n 


tr 




1 st term 

2 d term 

+557.1107 
+ 10.0559 

3d term 
4th term 

+0.0075 
-0.0449 




3d and 4th \ 
terms f 

+567.1666 

- 0.0374 

s 

sin a 

A' 

sec 

4_QO 

4. 745169 

9. 9780929 
8 . 5087447 
0. 2384490 

Arg. 

s 

J\ 


JX 

sin £(<£+<£') 
sec 

— J<j> 

+567.1292 

o t tr 

- 56 
+ 148 

2 (<£+<£') 

54 48 14. 3 


3. 4704648 

Corr. 

+ 92 




J\ 

+2954. 3694 



—Ja 


2. 7458 
9. 4465 
6. 4597 


8. 6520 


3.4704648 
9.9123203 


3. 3827851 

n 

+2414. 27 































































APPLICATION OF LEAST SQUARES TO TRL\NGULATION 


'primary triangulation —Continued 

STATION TOW HILL 


Third angle 


a 

Act 


<t> 

A<t> 

<y 


s 

COS a 

B 


1st term 
2d term 


3d and 4th \ 
terms 

— J<f> 


Tower to Lazaro 
Tow Hill and Lazaro 

Tower to Tow Hill 


Tow Hill to Tower 


54 


35 

31 


54 


4.870159 
9.8881303 
8. 5097404 


04 


27. 326 
01.527 


25.799 

-1 


3. 2680357 

n 

+1853. 6842 
+ 7.8777 


+ 1861.5619 
- 0.0351 


+ 1861.5268 

O / II 

54 19 56.6 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 
sec <£' 


JX 


Tower 


Tow Hill 


9. 74032 
9. 60486 
1. 55122 


0. 89640 

II 

+0. 0809 
-0.1160 


+ 16 
4. 870159 
9. 8024314 
8.5087606 
0. 2315526 




Arg. 

s 

JX 


3.4129052 I Corr. 

n 

+2587. 6482 


X 

JX 


6.5397 
2.3683 


8.9080 


- 98 
+ 114 


16 


o 

1 

II 

150 

32 

18.7 

-111 

09 

20.3 

39 

22 

58.4 

— 

35 

02.3 

180 



218 

47 

56.1 

131 

04 

48.015 

+ 

43 

07. 648 

131 

47 

55.663 



+ 2 

-h 

3. 2680 


s 2 sin 2 a 
E 


9.3452 
6. 4516 


9. 0648 


JX 

sin h(<f>+(f>') 
S3C %(J<f>) 


3. 4129052 
9. 9097770 
44 


—Ja 


j 3.3226866 

it 

+2102.3 


STATION NICHOLS 







o 

1 

II 

a 

Tow Hill to Lazaro 


197 

07 

18.4 

Third angle 

Nichols and Lazaro 


- 36 

08 

01.6 

a 

Tow Hill to Nichols 


160 

59 

16.8 

Ja 





— 

18 

56.0 






ISO 



a’ 

Nichols to Tow Hill 

0 

340 

40 

20.8 


O 

1 

II 







54 

04 

25. 798 

Tow Hill 

X 

131 

47 

55. 665 

J<f> 

+ 

39 

04. 894 


JX 

+ 

23 

17.123 


54 

43 

30. 692 

Nichols 

X' 

132 

11 

12. 788 


S 

COS a 

B 


1st term 
2d term 


3d and 4 th 1 
terms / 

— J<f> 


4.885142 
9.9756388 
8. 5097780 


3.3705588 

It 

-2347. 2470 
+ 2.1833 


-2345. 0637 
+ 0.1694 


-2344. 8943 

O t /1 

54 23 58. 9 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec <£' 


JX 


-1 

9. 77028 
9. 02582 
1. 54301 


-1 


0. 33911 

II 

+0.1292 
+0. 0402 


-70 
4.885142 
9. 5129060 
8. 50S7447 
0. 2384490 


3.1452347 

It 

+ 1397.1232 


(W 

D 


Arg. 

s 

JX 


Corr. 


6.7403 
2. 3709 


9.1112 


-104 
+ 34 


- 70 


—h 

s 2 sin 2 a 
E 


3.3706 
8. 7959 
6. 4373 

8. 6038 


JX 

sin 

sec £(J<jS) 


3.1452347 
9. 9101427 
70 


3. 0553844 

II 

-Ja +1136.02 


91865°—15-6 

































































80 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation, 


STATION KEN 








o 

/ 

n 

a 

Lazaro to Nichols 



71 

57 

14.5 

Second angle 

Nichols and Ken 



+ 22 

32 

58.8 

a. 

Lazaro to Ken 




94 

30 

13.3 

Ja 






— 

30 

53.7 







180 



a' 

Ken to Lazaro 




273 

59 

19.6 





First angle of triangle 

116 

59 

50.8 


O 

/ 

ft 






a 

54 

52 

57. 820 

Lazaro 

X 

131 

21 

58. 417 

j<f> 

+ 

1 

36. 965 


J\ 

+ 

37 

45. 909 


54 

54 

34. 785 

Ken 

X' 

131 

59 

44.326 


s 

cos a 

B 

4. 607407 

8.8949989 
8. 5097191 

S 2 

sin 2 a 

C 

9. 21480 

9. 99731 

1. 55589 

(W 

D 

4.0250 
2.3667 

-h 

s 2 sin 2 a 

E 

h 

2. 0121250 


0. 76800 


6.3917 



// 


// 




1st term 

2d term 

-102.8312 
+ 5.8614 

3d term 
4th term 

+0. 0002 
+0. 0048 




3d and 4th \ 
terms / 

- 96.9698 

+ 0.0050 

s 

sin a 

A' 

sec 4 V 

+58 
4.607407 

9. 9986569 
8. 5087403 
0. 2404325 

Arg. 

s 

JX 


JX 

sin h(4>+4>') 
sec $(A4>) 

— J4> 

- 96.9648 

o / n 

-29 

+87 

2 (<£ + <£') 

54 53 46. 4 


3. 3552425 

Corr. 

+58 




JX 

+2265.9094 



—Ja 


2.0125 
9. 2121 
6.4597 


7.6843 


3. 355242 
9. 912798 


3. 268040 
// 

+ 1853.70 


STATION ROUND 











O 


r 

ft 

a 

Lazaro and Ken 





94 

30 

13.3 

Second angle 

Ken and Round 





+ 78 

57 

17.0 

OL 

Lazaro to Round 





173 

27 

30.3 

Ja 









— 


i 

37.8 










180 




a' 

Round to Lazaro 





353 

25 

52.5 







First angle of triangle 

74 


42 

36.8 

(t> 

O 

54 


t 

52 

ft 

57. 820 


Lazaro 

X 

131 

21 

58. 417 

J<f> 

+ 


9 

58. 327 



JX 

+ 


1 

59. 499 

4>' 

55 


02 

56.147 


Round 

X' 

131 

23 

57. 916 













+ 1 

s 

4. 270074 


s 2 

8.54013 



-h 


2. 7769 

eos a 

9. 9971633 

sin 2 a 

8.11255 

(<W 2 

5.5538 

s 2 sin 2 

a 

6. 6527 

B 

8. 5097191 


C 

1. 55589 

D 

2. 3667 

E 


6. 4597 

h 

2. 7769564 


- 

8. 20857 


7. 9205 



5. 8893 

1st term 

-598. 3514 

3d term 

+0. 0083 







2d term 

+ 0.0161 

4 th term 

+0.0001 








-598. 3353 











3d and 4 th 1 

+ 0.0084 


s 

-6 

4. 270074 










sin a 

9. 0566163 

Arg. 


J A 


2. 077365 

— J4> 

-598. 3269 

A' 

8. 5087369 

s 

-6 

sin i(4>+4>') 

9. 913183 


O / 

ft 

sec 4> 

0. 2419389 

JX 

+0 

sec \{J4>) 



i(4>+4>') 

54 57 

57 



2. 0773655 

Corr. 

-6 



1. 990548 




J\ 

+ 119. 4993 



—Ja 

+97. 85 




































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


81 


■primary triangulation —Continued 

STATION KEN 


0 

251 
- 40 

f 

■ 17 
27 

ft 

00.2 

12.6 

210 

49 

47.6 

+ 

9 

22.7 

180 



30 

59 

10.3 
+ .1 

132 

11 

12. 787 

— 

11 

28. 462 

131 

59 

44. 325 
+ 1 

-h 

2 . 8226 


Third angle 


a 

Jet 


* 

j<f> 

V 


s 

COS a 

B 


1 st term 
2 d term 


3d and 4th \ 
terms / 

— d<j> 

i(<£+<£') 


Nichols to Lazaro 
Ken and Lazaro 

Nichols to Ken 


Ken to Nichols 


54 


+ 


43 

11 


54 


4.379021 
9.9338378 
8 . 5097307 


54 


30. 691 
04.093 


£4. 784 
+ 1 


2. 8225895 

ft 

-664. 6446 
+ 0.5380 

-664.1066 
+ 0.0132 


-664.0934 

O / // 

54 49 02.9 


S 2 

sin 2 a 
C 


3d term 
4th term 


sm a 
A' 

sec <f>' 


JX 


8 . 75804 

9. 41937 
1. 55337 


Nichols 

Ken 

(^) 2 


9. 73078 

ft 

+0.0103 
+0. 0029 

-2 

4. 379021 
9. 7096861 
8 . 5087403 
0. 2404325 


2. 8378797 

ft 

-688.4616 


Arg. 

s 

JX 


Corr. 


X 

JX 

X' 


5.6450 
2. 3676 


8.0126 


-10 
+ 8 


- 2 


s 2 sin 2 a 
E 


8.1775 
6. 4553 


7. 4554 


JX 

sin 

sec £(J<£) 


—da 


2 . 837880 
9. 912392 


2. 750272 

ft 

-562.69 


STATION ROUND 







O 

t 

ft 

a 

Ken to Lazaro 



273 

59 

19.6 

Third angle 

Round and Lazaro 


- 26 

20 

08.1 

CL 

Ken to Round 



247 

39 

11.5 

da 





+ 

29 

17.8 






180 



a ' 

Round to Ken 



68 

08 

29.3 


O 

t 

ft 






<t> 

54 

54 

34. 785 

Ken 

X 

131 

59 

44. 326 

d<{> 

+ 

8 

21. 362 


JX 

— 

35 

46. 407 


55 

02 

56.147 

Round 

X' 

131 

23 

57.919 









-2 


s 

COS a 

B 


1 st term 
2 d term 


3d and 4th 
terms 

— d<f> 
£(<£+<£') 


4. 614937 
9.5800256 
8. 5097172 


2. 7046798 

ft 

-506. 6171 
+ 5.2285 


-501.3886 
+ 0.0270 


-501. 3616 


54 58 46.0 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 
sec 


JX 


9. 22987 
9. 93219 
1. 55631 


0. 71837 

ft 

+0.0058 
+0.0212 

+49 
4. 614937 
9. 9660945 
8.50S7369 
0. 2419389 


3.3317122 

n 

-2146. 4074 


(<5g) 2 


Arg. 

5 

JX 


Corr. 


5.4091 
2.3666 


7. 7757 


-30 
+ 79 


+ 49 


-h 

s 2 sin 2 a 
E 


2. 7046 
9.1621 
6. 4604 


8 . 3271 


JX 

sin 

sec UH) 


—da 


3.331712 
9.913254 


3. 244966 

ff 

-1757. 79 

































































82 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation, 

STATION CAT 


o 

/ 

tt 

173 

27 

30.3 

+ 26 

22 

54.0 

199 

50 

24.3 

+ 

4 

21.4 

180 



19 

54 

45.7 

89 

47 

51.7 

131 

21 

58.417 

— 

5 

19. 289 

131 

16 

39.128 



+ 1 

-h 

2. 7062 

s 2 sin 2 a 

7. 5078 

E 

6. 4597 


Second angle 


Ja 


0 

J<j> 


s 

COS a 

B 


1st term 
2d term 


3d and 4th \ 
terms / 

— J<j> 

£( 0 + 0 ') 


Lazaro to Round 
Round and Cat 

Lazaro to Cat 


Cat to Lazaro 


54 


+ 


52 

8 


55 


4. 223072 
9.9734251 
8. 5097191 


01 


57. 820 
28.290 


26.110 


First angle of triangle 
Lazaro 
Cat 


2.7062162 


-508.4124 
+ 0.1158 


-508.2966 
+ 0.0065 


-508. 2901 

o t tt 

54 57 42.0 


s 2 

sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec 0' 


JX 


8.44612 
9. 06164 
1. 55589 


9.06365 


+0.0060 
+0.0005 


-3 
4.223072 
9.5307069 
8.5087375 
0. 2416677 


2.5041838 


-319. 2889 


Arg. 

s 

JX 


Corr. 


X 

JX 

X' 


5. 4124 
2. 3667 


7. 7791 


-5 

+2 


-3 


6.6737 


JX 

sin £(0+0') 
sec £C4<£) 


-Ja 


2. 504184 
9.913117 


2. 417301 


-261. 4 


STATION BEAVER 



* 








o 


t 

ft 

a 

Cat to Round 






109 

42 

37.4 

Second angle 

Round and Beaver 





+ 87 

34 

54.0 

a 

Cat to Beaver 






197 

17 

31.4 

Ja 









+ 


1 

40.2 










180 




a ' 

Beaver to Cat 






17 

19 

11.6 







First angle of triangle 

49 

58 

13. 2 


O 


t 


tt 








0 

55 


01 

26.110 


Cat 

X 

131 

16 

39.129 

J 0 

+ 


3 

45.204 



JX 

— 


2 

02. 231 

0' 

55 


05 

11.314 


Beaver 

X' 

131 

14 

36. 898 

s 

3.862988 


,s' 2 

7. 7260 



-h 


2.353 

COS a 

9. 9799133 

sin 2 a 

8.9465 

U0) 2 

4.705 

s 2 sin 2 

CL 

6. 672 

B 

8. 5097090 


C 

1. 5584 

D 

2.366 

E 


6. 464 

h 

2. 3526103 



8. 2309 


7.071 



5.489 


n 




tt 








1st term 

-225. 2218 

3d term 

+0.0012 







2d term 

+ 0.0170 

4 th term 










-225. 2048 











3d and 4th 1 

+ 0.0012 


s 

-1 

3. 862988 







leriiib j 



sin a 

9. 4731109 

Arg. 


JX 


2. 087181 

—J0 

-225.2036 

A' 

8. 5087360 

s 

-1 

sin £(0+0') 

9.913657 


O / 

tt 

sec 0' 

0. 2423463 

JX 

0 

sec £(J0) 



£(0+0') 

55 03 18. 7 



2.0871811 

Corr. 

-1 



2. 000838 




JX 

tt 

-122.2309 



—Ja 


tt 

-100.19 
































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


83 


'primary triangulation —Continued 

STATION CAT 







o 

/ 

n 

a 

Round to Lazaro 


353 

25 

52.5 

Third angle 

Cat and Lazaro 



- 63 

49 

14.7 

a 

Round to Cat 



289 

36 

37.8 

Ja 





+ 

5 

59.6 






180 



a ' 

Cat to Round 



109 

42 

37.4 


O 

/ 

it 






0 

55 

02 

56.147 

Round 

X 

131 

23 

57.917 


— 

1 

30. 037 


J\ 

— 

7 

18. 788 

V 

55 

01 

26.110 

Cat 

X' 

131 

16 

39.129 


s 

COS a 

13 

h 


1st term 
2d term 


3d and 4th \ 
terms 

— J<f> 


3.917800 
9.5268533 
8. 5097071 


1.9533604 


+89. 8174 
+ 0.2199 


+90.0373 
0 


+90. 0373 

O tit 

55 02 11.1 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 
sec 


J\ 


7.8356 
9.9481 
1. 5586 


9.3423 


+ 0.0002 
- 0.0002 


+2 
3. 917800 
9. 9740491 
8. 5087375 
0.2416677 


2. 6422545 

n 

-438. 7877 


(<^) 2 


Arg. 

s 

J\ 


Corr. 


3.9069 
2.3658 


6. 2727 


-1 

+3 


+2 


-h 

s 2 sin 2 a 
E 


1.9535 
7. 7837 
6. 4643 


6. 2015 


J\ 

sin %(<!>+<{>') 
sec h(W) 


—Ja 


2. 642254 
9.913558 


2.555812 

tt 

-359. 59 


STATION BEAVER 







o 

/ 

it 

a 

Round to Cat 



289 

36 

37.8 

Third angle 

Beaver and Cat 



- 42 

26 

53.0 

a 

Round to Beaver 


247 

09 

44.8 

Ja 





+ 

7 

39.9 






180 



a' 

Beaver to Round 


67 

17 

24.7 








+.1 


O 

/ 

ft 






<t> 

55 

02 

56.147 

Round 

X 

* 131 

23 

57. 917 

J<f> 

+ 

2 

15.166 


J\ 

— 

9 

21.019 


55 

05 

11. 313 

Beaver 

X' 

131 

14 

36. 898 




+ 1 






s 

COS a 

B 

4.033348 

9.5889657 
8 . 5097071 

S 2 

sin 2 a 

C 

8.0667 

9. 9291 
1.5586 

(W 

4.264 

2.366 

-h 

s 2 sin 2 a 
E 

h 

2.1320208 


9.5544 


6.630 



n 


// 




1 st term 

2 d term 

-135.5254 
+ 0.3584 

3d term 
4th term 

+0.0004 

+0.0004 





-135.1670 


+3 
4.033348 

9. 9645467 
8 . 5087360 
0. 2423463 




3d and 4 th 1 
terms / 

+ 0.0008 

s 

sin a 

A' 

sec <f>' 

Arg. 

s 

J\ 


JX 

sin $(<t>+<f> 
sec h(J<f>) 

— J<f> 

-135.1662 

O / // 

-2 

+5 

£(<£+<£') 

55 04 03. 7 

J\ 

2. 7489773 

it 

-561.0186 

Corr. 

+3 

—Ja 


2.132- 
7. 996 
6. 464 


6.592 


2. 748977 
9.913723 


2. 662700 


-459. 94 






























































COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final 'position computation, 


STATION LIM 











O 

t 

rr 

a 

Beaver to Cat 






17 

19 

11.6 

Second angle 

Cat and Lim 






+ 102 

39 

46.2 

a 

Beaver to Lim 






119 

58 

57.8 

Jot 









— 

5 

20.3 










180 



a ' 

Lim to 

Beaver 






299 

53 

37.5 







First angle of triangle 

36 

34 

56.6 


O 


t 


tt 







<t> 

55 


05 

11.314 

Beaver 

X 

131 

14 

36. 898 

j<j> 

+ 


2 

08.948 



J\ 

+ 

6 

30. 508 

<t>' 

55 


07 

20.262 


Lim 

X' 

131 

21 

07* 406 












+ 1 

s 

3.902553 

S 2 

7.8051 



—h 

2.111 

COS a 

9. 6987431 

sin 2 a 

9. 8751 

(M) 2 

4. 222 

s 2 sin 2 a 

7. 680 

B 

8. 5097044 


C 

1. 5591 

D 

2.366 

E 

6. 465 

h 

2.1110005 



9. 2393 


6.588 


6. 256 


// 




tt 







1st terra 

-129.1221 

3d term 

+0.0004 






2d term 

0.1735 

4th terra 

+0.0002 







-128.9486 





* 





3d and 4 th 1 

+ 0.0006 


.9 

+ 2 

3.902553 






terras / 



sin a 

9.9376062 

Arg. 


J\ 

2. 591630 

—J4> 

-128.9480 

J 

V 

8. 5087351 

s 

-1 

sin J(0+<£ , ) 

9. 913918 




sec <£' 

0. 2427356 

J\ 

+3 

sec $(J$) 




O / 

tt 











55 06 15. 8 



2. 5916301 

Corr. 

+2 


2. 505548 







tt 





ft 




J\ 

+390. 5082 



—Aa 


+320. 3 


STATION SOUTH TWIN 


• 






O 

t 

tt 

a 

Beaver to Lim 




119 

58 

57.8 

Second angle 

Lim and South Twin 



+ 78 

24 

37.0 

a 

Beaver to South Twin 



198 

23 

34. 8 

Ja 






+ 

1 

27.7 







180 



a' 

South Twin to Beaver 



18 

25 

02.5 





First angle of triangle 

60 

40 

04. 7 


o 

t 

tt 






4> 

55 

05 

11. 314 

Beaver 

X 

131 

14 

36. 898 

j(f> 

+ 

3 

04. 203 


JX 

— 

1 

46. 925 


55 

08 

15. 517 

South Twin 

X' 

131 

12 

49. 973 









+1 




15.517 





49. 974 


s 

COS a 

B 

3. 778398 

9.9772271 
8. 5097044 

« 2 

sin 2 a 

C 

7. 5568 

8.9984 

1. 5591 

(<g ) 2 

4.530 
2.366 

-h 

s 2 sin 2 a 

E 

h 

2. 2653295 


8.1143 


6.896 


1st term 

2d term 

// 

-184. 2169 
+ 0.0130 

3d term 
4th term 

tt 

+0.0008 




3d and 4 th \ 
terms / 

-184. 2039 

+ 0.0008 

s 

sin a 

A' 

sec <f>' 

-1 
3. 778398 

9. 4990449 
8. 5087347 
0. 2429024 

Arg. 

s 

J\ 

• 

J\ 

sin h(<t>+4>') 
sec 

— J<f> 

-184.2031 

O t ft 

-1 

0 


55 06 43. 4 


2. 0290799 

Corr. 

-1 




J\ 

ft 

-106.9252 



—Ja 


2.265 
6. 555 
6. 465 


5. 285 


2.029080 
9. 913958 


1. 943038 
// 

-87.7 































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


85 


;primary triangulation —Continued 


STATION LIM 











0 


f 

tt 

a 

Cat to Beaver 






197 

17 

31. 4 

Third angle 

Lim and Beaver 





— 40 

45 

17.3 

a 

Cat to Lim 







156 

32 

14.1 

Ja 









— 


3 

40.0 










180 




a' 

Lim to Cat 







336 

28 

34.1 


O 


f 


n 








0 

55 


01 

26.110 


Cat 

X 

131 

16 

39. 129 

a <j> 

+ 


5 

54.152 



AX 

+ 


4 

28. 278 

0' 

55 


07 

20.262 


Lira 

X' 

131 

21 

07. 407 

s 

4.077064 


$2 

8.1541 



-h 


2.549 

COS a 

9.9625205 

Sin 2 a 

9.1999 

(#) 2 

5.099 

s 2 sin 2 a 

7. 354 

B 

8. 5097090 


C 

1.5584 

D 

2.366 

E 


6. 464 

h 

2. 5492935 



8.9124 


7.465 



6.367 


tt 




n 








1st term 

-354. 2367 

3d term 

+0. 0029 







2d term 

+ 0.0817 

4th term 

+0.0002 








-354.1550 











3d and 4th \ 

+ 0.0031 


s 

4.07 

7064 










sin a 

9. 6000497 

Arg. 


AX 


2. 428584 

-A$ 

-354.1519 


A' 

8. 5087351 

s 

-2 

sin %(4>+<f>') 

9. 913752 


O / 

n 

sec <j) 

0. 2427356 

A\ 

+2 

sec %(A<t>) 




55 04 23. 2 



2. 4285844 

Corr. 

0 



2. 342336 







tt 






tt 




AX 

+268. 2776 



—Aot 

+219.96 


STATION SOUTH TWIN 











o 


t 

tt 

a. 

Lim to Beaver 






299 

53 

37.5 

Third angle 

South Twin and Beaver 




- 40 

55 

18.4 

a 

Lim to South Twin 





258 

58 

19. 1 

Aot 









+ 


6 

48.1 










180 




a' 

South Twin to Lim 





79 

05 

07.2 

<i> 

O 

55 


/ 

07 

tt 

20.262 


Lim 

X 

131 

21 

07. 407 

A<t> 

+ 



55. 255 



AX 

— 


8 

17. 433 

4>' 

55 


08 

15. 517 

South Twin 

X' 

131 

12 

49. 974 

s 

3.953192 

s 2 

7.9064 



-h 


1.744 

COS a 

9. 2816903 

sin 2 a 

9. 9838 

(W 

3. 488 

s 2 sin 2 a 

7. 890 

B 

8. 5097018 


C 

1. 5597 

D 

2.365 

E 


6. 467 

h 

1. 7445841 



9.4499 


5.853 



6.101 


tt 




ft 








1st term 

-55. 5372 

3d term 

+0.0001 







2d term 

+ 0. 2818 

4th term 

+0.0001 








-55. 2554 











3d and 4th \ 

+ 0.0002 


$ 

+z 
3. 953192 







terms / 



sin a 

9. 9919052 

Arg. 


AX 


2. 696734 

— A<f> 

—55. 2552 


V' 

8. 5087347 

s 

-2 

sin K<£+<£') 

9. 914053 



• 

sec 4>' 

0. 2429024 

AX 

+4 

sec i(A<t>) 




O / 

// 











£(<£+<£')' 

55 07 47.9 



2. 6967345 

Corr. 

+2 



2.610787 







tt 






tt 




AX 

-497. 4329 



—Act 


-408.12 
































































86 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final 'position computation , 


( 


STATION SEAL 








0 

t 

ft 

a 

Nichols to Ken 




210 

49 

47.6 

Second angle 

Ken and Seal 




+ 27 

51 

41.5 

a 

Nichols to Seal 




238 

41 

29.1 

Aa 






+ 

29 

04. 7 







180 



ct r 

Seal to Nichols 




59 

10 

33.8 









+ • 1 





First angle of triangle 

25 

37 

10.6 


O 

t 

ft 






4> 

54 

43 

30. 691 

Nichols 

X 

132 

11 

12. 787 

A<J> 

+ 

12 

22.365 


JX 

— 

35 

34.411 

F 

54 

55 

53.056 

Seal 

X' 

131 

35 

38.376 




-1 





+ 1 

s 

4.648215 

s 2 9.2964 


-h 

2. 874 


COS a 

B 


1st term 
2d term 


3d and 4th 1 
terms / 

— A<j> 


9. 7157086 
8. 5097307 


2.8736543 


-747. 5741 
+ 5.1654 


-742. 4087 
+ 0.0438 


-742.3649 

o t rt 

54 49 41.9 


sin 2 a 
C 


3d term 
4th term 


s 

sin a 
A' 

sec F 


JX 


9.8633 
1.5534 


0. 7131 


+0.0130 
+0. 0308 


+4 
4.648215 
9.931652 
8.508740 
0. 240667 


3. 329278 

ft 

-2134. 4108 


1 ) 


Arg. 

s 

JX 


Corr. 


5.747 
2. 368 


8.115 


-4 

+8 


+4 


s 2 sin 2 a 
E 


9.160 
6. 455 


8. 489 


JX 

sin 

sec §(J0) 


—Aa 


3.329278 
9.912450 


3. 241728 

ft 

-1744. 72 


STATION MID 











O 


/ 

tt 

a 

Round to Lazaro 





353 

25 

52.5 

Second angle 

Lazaro and Mid 





+ 51 

05 

14.8 

a 

Round to Mid 






44 

31 

07.3 

Aa 









— 


6 

56.5 










180 




a' 

Mid to Round 






224 

24 

10.8 







First angle of triangle 

85 

15 

12.2 


O 


f 


ft 








<t> 

55 


02 

56.147 

Round 

X 

131 

23 

57.917 

A<}> 

— 


4 

57. 777 



JX 

+ 


8 

28. 436 

<t>' 

54 


57 

58. 370 


Mid 

X' 

131 

32 

26. 353 

-1 

s 

4.110647 


s 2 

8.2213 



-h 


2.473 

cos a 

9. 853103 

sin 2 a 

9. 6916 

(W 

4.947 

s 2 sin 2 a 

7.913 

B 

8. 509707 


C 

1.5586 

D 

2.366 

E 


6. 464 

4 

h 

2. 473457 



9. 4715 


7.313 



6.850 

1st term 

ft 

+297. 4795 

3d term 

tf 

+0.0021 







2d term 

+ 0.2961 

4th term 

-0. 0007 








+ 297.7756 











3d and 4 th 1 
terms / 

+ 0.0014 


s 

4.110647 










9. 845806 

Arg. 


JX 











Z./UUZOO 

— A<j> 

+297. 7770 

j 

V 

8.508740 

s 


sin 

9. 913405 


O / 

ft 

sec <j>' 

0. 241043 

JA 


sec 




; 55 00 27.3 



2. 706236 

Corr. 




2. 619641 




JX 

tf 

+508. 4356 



—Aa 

ft 

+416.52 

































































APPLICATION OF LEAST SQUALLS TO TRIANGTJLATION 


primary triangulation —Continued 

STATION SEAL 















o 

/ 

rr 

a 

Ken to Nichols 



. 30 

59 

10. 4 

Third angle 

Seal and Nichols 


-126 

31 

09. 1 

a 

Ken to Seal 



264 

28 

01.3 

da 





+ 

19 

43.3 






180 



a' 

Seal to Ken 



84 

47 

44.6 








-.1 


O 

r 

rr 







54 

54 

34. 785 

Ken 

X 

131 

59 

44. 326 

J<t> 

+ 

1 

18. 270 


JX 

— 

24 

05. 949 

<t>' 

54 

55 

53.055 

Seal 

X' 

131 

35 

38. 377 


s 

cos a 
B 


1 st term 
2 d term 


3d and 4th \ 
terms / 

— A4> 
£(<£+<£') 


4.412771 
8.984161 
8 . 509717 


1.906649 


-80.6583 
+ 2.3867 


-78. 2716 
+ 0.0017 


-78. 2699 

or rr 

54 55 13.9 


s 2 

sin 2 « 

C 


3d term 
4th term 


s 

sin a 
A' 

sec <j>' 


JX 


8.8255 
9.9960 
1. 5563 


0. 3778 


+0.0002 
+0.0015 


+3 
4. 412771 
9.997972 
8.508740 
0. 240667 


3.160153 

rr 

-1445.949 


(4g)* 


Arg. 

s 

JX 


Corr. 


3.813 
2.367 


6.180 


-1 
+ 4 


+3 


-h 

A 2 sin 2 a 
E 


1.907 
8 . 821 
6 . 460 


7.188 


JX 

sin 

sec £(J<A) 


-da 


3.100153 
9.912942 


3. 073095 

rr 

-1183.30 


STATION MID 







0 

r 

rr 

a 

Lazaro to Round 


173 

27 

30.3 

Third angle 

Mid and Round 


- 43 

39 

33.5 

a 

Lazaro to Mid 



129 

47 

56.8 

da 





— 

8 

33.9 






180 



a' 

Mid to Lazaro 



309 

39 

22.9 








+ .1 


O 

/ 

rr 






(f> 

54 

52 

57. 820 

Lazaro 

X 

131 

21 

58. 417 

d<f> 

+ 

5 

00. 550 


JX 

+ 

10 

27. 934 


54 

57 

58.370 

Mid 

X' 

131 

32 

26. 351 









+ 1 


s 

COS a 

B 

4.162604 
9. 806246 
8 . 509719 

S 2 

sin 2 a 

C 

8 . 3252 

9. 7710 

1. 5559 

(W) 1 

4.957 
2.367 

-h 

s 2 sin 2 a 

E 

2. 479 

8 . 096 

6 . 460 

h 

2. 478569 


9. 6521 


7.324 


7.035 

1 st term 

2 d term 

rr 

-301.0017 
+ 0.4488 

3d term 
4th term 

rr 

+0.0021 
+0.0011 





3d and 4th 1 
terms / 

-300. 5529 

+ 0.0032 

s 

sin a 

A' 

sec <j>' 

4.162604 
9.885527 
8 . 508740 
0. 241043 

Arg. 

s 

JX 


JX 

sin i(^&+<£ , ) 
sec i(J<£) 

2. 797914 

9.912963 

— d(f> 

-300.5497 

or rr 



54 55 28.1 


2. 797914 

Corr. 



2. 710877 



d\ 

rr 

+627.9340 



—da 

rr 

+513.90 
































































88 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 23 


Final position computation, 


STATION SPUR 











O 


r 

ft 

a 

Round 1 to Cat 






289 


36 

37.8 

Second angle 

Cat and Spur 






+ 111 

30 

08.5 

a 

Round to Spur 






41 

06 

46.3 

Ja 









— 


4 

22. 7 










180 




a' 

Spur to Round 






221 

02 

23.6 







First angle of triangle 

33 

20 

38.3 


0 


t 


ft 








<t> 

55 


02 

56.147 

Round 

X 

131 

23 

57. 917 

A<j> 

- 


3 

31.319 



AX 

+ 


5 

20. 567 

F 

54 


59 

24. 828 
-1 


Spur 

X' 

131 

29 

18. 484 

s 

3.937953 


s 2 

7. 8759 



-h 



2.325 

COS a 

9.877035 

sin 2 a 

9. 6358 

(<J$) 2 

4.649 

s 2 sin 2 

a. 


7. 512 

B 

8. 509707 


C 

1. 5586 

D 

2.366 

E 



6.464 

h 

2.324695 



9. 0703 


7.015 




6.301 

1st term 

n 

+211.2006 

3d term 

n 

+0. 0010 







2d term 

+ 0.1176 

4th term 

-0.0002 








+211.3182 











3d and 4th \ 

+ 0.0008 


s 

3. 937953 







t>61 ills J 



sin a 

9.817925 

Arg. 


AX 


2. 505919 

— A4> 

+211.3190 

A' 

8. 508738 

s 


sin £(<£+<£') 

9.913468 


O t 

ft 

sec <])' 

0. 241303 

AX 


sec %(A<f>) 




55 01 10. 5 



2. 505919 

Corr. 




2. 419387 






/ 







ft 




AX 

+320.5672 



—Aoc 

+262.66 


STATION SNIPE 











O 


t 

if 

a 

Round to Cat 






289 

36 

37.8 

Second angle 

Cat and Snipe 






+ 63 

10 

27.2 

a 

Round to Snipe 





352 

47 

05.0 

Aa 









+ 



29.8 










180 




a' 

Snipe to Round 





172 

47 

34.8 







First angle of triangle 

79 

10 

09.9 


O 


t 


ft 








4> 

55 


02 

56.147 

Round 

X 

131 

23 

57. 917 

A<j> 

— 


2 

45.140 



AX 

— 



36. 371 

<f>' 

55 


00 

11.007 


Snipe 

X' 

131 

23 

21. 546 

s 

3. 711593 


s 2 

7. 4232 



-h 


2. 218 

COS a 

9. 996547 

sin 2 a 

8.1980 


4. 436 

s 2 sin 2 

a 

5. 621 

B 

8. 509707 


C 

1. 5586 

D 

2. 366 

E 


6. 464 

h 

2. 217847 



7.1798 


6. 802 



4.303 

1st term 

ft 

+ 165.1380 

3d term 

ft 

+0.0006 







2d term 

+ 0.0015 

4th term 










+ 165.1395 











3d and 4 th \ 

+ 0.0006 


s 

3. 711593 










sm a. 

9.098982 

Arg. 


AX 


1. 560755 

— A<f> 

+ 165.1401 


a' 

8. 508738 

s 


sin §(<£+<£') 

9. 913502 


O / 

ft 

sec 

0.241442 

A\ 


sec %(A<f>) 




55 01 33.6 



1. 560755 

Corr. 




1. 474257 




A\ 

ft 

-36.3710 



—Act 


ft 

-29.80 

































































APPLICATION OF 1 LEAST SQUARES TO TRIANGULATION 


89 


primary triangulation —Continued 


STATION SPUR 



o 

109 
- 35 

1 

42 

09 

tt 

37.4 

13. 4 


74 

33 

24. 0 


— 

10 

22. 1 


180 




254 

23 

01.9 

X 

131 

16 

39.129 

JX 

+ 

12 

39. 355 

X' 

131 

29 

18. 484 


Third angle 


a 

Jot 


<f> 

j <{> 

<f>' 


Cat to Round 
Spur and Round 

Cat to Spur 


Spur to Cat 


55 


54 


01 

2 


59 


26.110 

01.283 


24. 827 


Cat 

Spur 


S 

COS a 

B 

4.146374 
9. 425347 
8. 509709 

S 2 

sin 2 a 

C 

8.2927 

9.9681 

1. 5582 

(<^) 2 

4.163 
2.366 

-h 

$ 2 sin 2 a 

E 

h 

2.081430 
/1 


9.8190 

// 


6.529 


1st term 

2d term 

+ 120.6230 
+ 0.6592 

3d term 
4th term 

+0.0003 

-0.0006 





+ 121.2832 


+1 

4.146374 
9.984029 
8. 508738 
0.241303 




3d and 4th \ 
terms / 

-0.0003 

s 

sin a 

A' 

sec <f>’ 

Arg. 

s 

J\ 


J\ 

sin i(<f>+<f>') 
sec j(J<f>) 

— J<f> 

+ 121.2829 

o / n 

+ 1 


55 00 25.5 


2. 880445 

Corr. 

+ 1 




J\ 

// 

+ 759.3552 



—Jot 


2.081 
8. 261 
6. 464 


6.806 


2. 880445 
9.913402 


2. 793847 

it 

-F622.07 


STATION SNIPE 











O 


I 

It 

CL 

Cat to Round 






109 

42 

37.4 

Third angle 

Snipe and Round 





- 37 

39 

23.0 

a 

Cat to Snipe 






72 

03 

14. 4 

Jot 









— 


5 

29. 7 










180 




ct' 

Snipe to Cat 






251 

57 

44.7 


O 


t 


// 








tf> 

55 


01 

26.110 


Cat 

X 

131 

16 

39. 129 

J<f> 

— 


1 

15.103 



J\ 

+ 


6 

42.417 


55 


00 

11.007 


Snipe 

X' 

131 

23 

21. 546 

S 

3. 876157 


s 2 

7. 7523 



-h 


1.975 

COS a 

9. 488721 

sin 2 a 

9. 9567 

(W)* 

3.949 

s 2 sin 2 a 

7. 709 

B 

8. 509709 


C 

1.5582 

D 

2. 366 

E 


6. 464 

h 

1.874587 



9. 2672 

. 

6.315 



6.148 


// 




It 







1st term 

+74.9181 

3d term 

+0.0002 







2d term 

+ 0.1850 

4th term 

-0. 0001 








+75.1031 











3d and 4 th \ 

+ 0.0001 


s 

3. 876157 







terms / 



sin a 

9. 978339 

Arg. 


J\ 


2.604676 

— Jtj> 

+ 75.1032 



8. 508738 

s 


sin £(<£+<£') 

9. 913436 




sec 6' 

0. 241442 

J\ 


sec £U<£) 




O / 

// 











£($+<£') 

55 00 48.6 



2. 604676 

Corr. 




2. 518112 




Jk 

It 

+ 402.4167 



—Jot 

+329. 69 































































90 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation , primary triangulation —Continued 

STATION KHWAIN 


Second angle 


Act 


<t> 

Atf> 


Beaver to Cat 
Cat and Khwain 

Beaver to Khwain 


Khwain to Beaver 


55 


55 


05 


04 


11.314 

55.322 


15.992 
-1 


First angle of triangle 
Beaver 


Khwain 


X 

AX 

X' 


o 

/ 

ft 

17 

19 

11.6 

+58 

43 

19.0 

76 

02 

30.6 

— 

5 

17.1 

ISO 



255 

57 

13.5 

62 

13 

31.4 

131 

14 

36.898 

+ 

6 

26.681 

131 

21 

03.579 


s 

cos « 

B 

3. 849454 

9.3824014 
8.5097044 

s~ 

sin 2 a 

C 

7.6989 

9.9740 

1.5591 

(W 

f) 

3.483 
2.366 

—h 

,s' 2 sin 2 a 
E 

h 

1. 7415598 


9. 2320 


5.849 


1 st term 

2 d term 

tt 

+55.1518 
+ 0.1706 

3d term 
4th term 

ft 

+0.0001 
-0.0001 



. 

3d and 4 th 1 
terms / 

+55.3224 

0.0000 

s 

sin a 

A' 

sec <j>' 

3. 849454 
9.986983 
8.508736 
0. 242180 

Arg. 

s 

AX 


AX 

sin h(4>+4> 
sec i(A<t>) 

— A<f> 

+55.3224 

Of tt 



55 04 43. 7 


2.587353 

Corr. 





AX 

tt 

+386.6812 



—Aa. 


1.742 
7.673 
6 . 465 

5.880 


2.587353 
9. 913782 


2. 501135 

tt 

+317.06 


STATION KHWAIN* 







O 

r 

tt 

a 

Cat to Beaver 



197 

17 

31.4 

Third angle 

Khwain and Beaver 


- 59 

03 

09.7 

a 

Cat to Khwain 



138 

14 

21.7 

Aa 





— 

3 

36.8 






ISO 



a ’ 

Khwain to Cat 



318 

10 

44.9 


O 

9 

99 






<t> 

55 

01 

26.110 

Cat 

X 

131 

16 

39.129 

Atf> 

+ 

2 

49.881 


A\ 

+ 

4 

24. 450 

<£' 

55 

04 

15.991 

IChwain 

X' 

131 

21 

03.579 


s 

COS a 

B 

3.847941 

9 872700 
8 . 509709 

S 2 

sin 2 a 

C . 

7.6959 

9.6470 

1.5584 

T 

4.461 
2.366 

-h 

s 2 sin 2 a 

E 

h 

2. 230350 


8.9013 


6.827 


1 st term 

2 d term 

tt 

-169.9613 
+ 0.0797 

3d term 
4th term 

tt 

+0.0007 
+0.0001 




3d and 4 th 1 
terms J 

-169.8816 

+ 0.0008 

s 

sin a 

A' 

sec <},' 

3. 847941 
9.823487 
8.508736 
0. 242180 

Arg. 

s 

AX 


AX 

sin §(<£+<£') 
sec j(J<£) 

— At}> 

-169.8808 

Of ft 


h(<f>+4>') 

55 02 51.1 


2. 422344 

Corr. 





AX 

ft , 

+264. 4502 



—Aa 


2.230 
7.343 
6. 464 


6.037 


2. 422344 
9.913616 


2.335960 

it 

+216.75 


* This is right-hand portion of computation above. 



































































APPLICATION OF LEAST SQLTARES TO TRIANGULATION. 
List of geographic positions—Felice Strait, Alaska, southeast Alaska datum 


91 


Station 

Latitude 

and 

longitude 

Sec¬ 
onds in 
meters 

Azimuth 

Back 

azimuth 

To station 

Distance 

Loga¬ 

rithm 


o 

/ 

n 


O 

f 

// 

O 

/ 

// 


Meters 


Tower 

54 

35 

27.326 

845.0 

201 

43 

27.2 

21 

50 

34.8 

Turn 

25288. 4 

4.402921 

1907 

131 

04 

48.015 

862.3 

243 

43 

58.0 

63 

51 

41.0 

Dundas 

11361.1 

4.055419 

Lazaro 

54 

52 

57.820 

1788.0 

287 

48 

01.7 

108 

09 

12.5 

Turn 

29163.6 

4.464841 

1907 

131 

21 

58.417 

1041.5 

313 

40 

37.9 

134 

02 

23.3 

Dundas 

39641.7 

4.598152 






330 

18 

17.4 

150 

32 

18.7 

Tower 

37351.0 

4.572302 

Tow Hill 

54 

04 

25.798 

797.6 

197 

07 

18.4 

17 

28 

25.9 

Lazaro 

94307.0 

4.974544 

1908 

131 

47 

55.665 

1012.2 

218 

47 

56.1 

39 

22 

58.4 

Tower 

74158. 2 

4.870159 

Nichols 

54 

43 

30.691 

949.0 

251 

17 

00.2 

71 

57 

14.5 

Lazaro 

55612.1 

4.745169 

1907 

132 

11 

12.787 

228.9 

281 

21 

55.6 

102 

16 

06.1 

Tower 

72983.6 

4.863225 






340 

40 

20.8 

160 

59 

16.8 

Tow Hill 

76761.2 

4.885142 

Ken 

54 

54 

34.785 

1075.7 

273 

59 

19.6 

94 

30 

13.3 

Lazaro 

40495.5 

4.607407 

1907 

131 

59 

44.326 

789.7 

30 

59 

10.4 

210 

49 

47.6 

Nichols 

23934.3 

4.379021 

Seal 

54 

55 

53.055 

1640.6 

290 

15 

23.0 

110 

26 

33.9 

Lazaro 

15582.6 

4.192639 

1907 

131 

35 

38.377 

683.4 

59 

10 

33.9 

238 

41 

29.1 

Nichols 

44485.1 

4.648215 






84 

47 

44.5 

264 

28 

01.3 

Ken 

25868.5 

4.412771 

Mid 

54 

57 

58.370 

1805.0 

309 

39 

23.0 

129 

47 

56.8 

Lazaro 

14541.3 

4.162604 

1914 

131 

32 

26.352 

468.8 

78 

00 

16.4 

257 

37 

55.6 

Ken 

29834.6 

4.474720 

Round 

55 

02 

56.147 

1736.3 

353 

25 

52.5 

173 

27 

30.3 

Lazaro 

18624.0 

4.270074 

1914 

131 

23 

57.917 

1028.3 

44 

31 

07.3 

224 

24 

10.8 

Mid 

12901. 7 

4.110647 






68 

08 

29.3 

247 

39 

11.5 

Ken 

41203.8 

4.614937 

Spur 

54 

59 

24.827 

767.7 

221 

02 

23.6 

41 

06 

46.3 

Round 

8668 . 7 

3.937953 

. 1914 

131 

29 

18.484 

328.7 

326 

44 

11.7 

146 

50 

11.9 

Lazaro 

14304.3 

4.155468 

Cat 

55 

01 

26.110 

807.4 

19 

54 

45.7 

199 

50 

24.3 

Lazaro 

16713. 7 

4.223072 

1914 

131 

16 

39.129 

695.2 

74 

33 

24.0 

254 

23 

01.9 

Spur 

14007. 9 

4.146374 






109 

42 

37.4 

289 

36 

37.8 

Round 

8275.6 

3.917800 

Snipe 

55 

00 

11.007 

340.4 

172 

47 

34.8 

352 

47 

05.0 

Round 

5147.5 

3. 711593 

1914 

131 

23 

21.546 

383.0 

251 

57 

44.7 

72 

03 

14.4 

Cat 

7518.9 

3.876157 

Beaver 

55 

05 

11.314 

349.9 

17 

19 

11.6 

197 

17 

31.4 

Cat 

7294.4 

3.862988 

1914 

131 

14 

36.898 

654.5 

45 

09 

02.7 

225 

01 

52.7 

Snipe 

13154.2 

4.119063 






67 

17 

24.8 

247 

09 

44.8 

Round 

10798.1 

4.033348 






148 

12 

30.2 

328 

08 

19.3 

Ham 

10275.3 

4.011793 






198 

23 

34.8 

18 

25 

02.5 

South Twin 

6003.4 

3. 778398 

Khwain 

55 

04 

15.991 

494.5 

255 

57 

13.5 

76 

02 

30.6 

Beaver 

7070.6 

3.849454 

1914 

131 

21 

03.579 

63.5 

318 

10 

44.9 

138 

14 

21.7 

Cat 

7046.0 

3.847941 

Lim 

55 

07 

20.262 

626.6 

197 

33 

05.0 

17 

34 

14.5 

Ham 

4974.6 

3.696759 

1914 

131 

21 

07.407 

131.3 

258 

58 

19.1 

79 

05 

07.2 

South Twin 

8978.3 

3.953192 






299 

53 

37.5 

119 

58 

57.8 

Beaver 

7990.1 

3.902553 






336 

28 

34.1 

156 

32 

14.1 

Cat 

11941.6 

4.077064 






359 

19 

01.9 

179 

19 

05.0 

Khwain 

5698. 8 

3. 755783 


ADJUSTMENT OF TRIANGULATION BY THE METHOD OF VARIATION 

OF GEOGRAPHIC COORDINATES 


DEVELOPMENT OF FORMULAS 

A scheme of triangulation may be adjusted not only by means of 
equations of condition * but also by means of observation equations 
in which the number of independent unknowns is just sufficient to 

* There is some confusion in usage as to the term equation of condition, or condition equation. In this 
publication the meaning is restricted to that of an equation expressing some condition which is imposed 
a priori and independently of anything arising from the observations themselves, and which must be 
rigorously satisfied by the adopted results. An equation which expresses the results of an observation, 
and which will, in general, be satisfied only approximately by the adopted results, is not herein termed an 
equation of condition, but an observation equation. 























92 COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 28. 

determine the entire triangulation. These independent unknowns 
may very conveniently be taken as the small corrections to the 
assumed approximate geographic coordinates (that is, the latitudes 
and longitudes) of the points in the triangulation. To form the 
observation equations the relation must be found that connects the 
small change in the direction of a line with the small arbitrary 
changes in the geographic coordinates of its ends. The following 
derivation of the formulas is based on the formulas for the compu¬ 
tation of geographic positions given in U. S. Coast and Geodetic 
Survey Special Publication No. 8 and on the notation there used. 
A “ d” before the symbol of a quantity denotes a small arbitrary 
change in that quantity. <j> and X are, respectively, the latitude and 
longitude of A v the initial point of the position computation, which 
may also be thought of as the occupied point, while 4>' and X ' are the 
latitude and longitude of B v the terminal point in the position com¬ 
putation, which may also be thought of as the point sighted on. By 
definition also, 

Acf> = 4>'-4> 

AX = X'-X 
h = sB cos a 

a is the azimuth at A t of the line A 1 B 1 reckoned from the south 
toward the west. 


AX = sA' sec (/>' sin a 
l 

COS Oi = — Ti 

sB 


sin 


AX 

sA' sec 4 >' 


cot a = 


A' sec 4>' 

~~B~~ 


A 

AX 


The meaning of A' and B is explained in Special Publication No. 8. 

By differentiating the preceding equation and neglecting the effects 
of changes in A', B, and sec 4>' there results: 


— cosec 2 a da = 


A' 


sec <j)TAXdh — Jid(AX)~] 

B L W 2 J 


(Axy 

s 2 A' 2 sec 2 4 V 


Multiplying by — sin 2 a = 







APPLICATION OF LEAST SQUARES TO TRIANGULATION. 93 

and dividing by arc 1" in order to express da in seconds instead of in 
radians gives, 

da in seconds-. ^, 

sB cos a sA' sin a sec <j>' 

s 2 BA' sec 0' arc ¥BA' sec # arc V ,8h 

= _ sin or cos a jn\ sin a cos a ^ 

sA' sec <f>' sin a arc 1"“' ' sB cos or arc 

= sin a cos af ~ d(AX) _ 

~ arc 1" l AX ~ hj 

By neglecting the variations in all the terms of the expression given 
for J(j> in Special Publication No. 8 except the first or principal term, 
Jij there results, 

d(d<t>) = —dJi = d(f)' — d(f) 

Evidently, also, 

d(AX)=dX'-dk 

It thus appears that, to the degree of approximation here adopted, 
it is the difference in the changes of coordinates at the ends of a line 
that turns the line in azimuth. The formulas for computing da 
become, 

da in SCC ' = ^' seeV arc W + W^-.M)] 

sin a cos a V dcj)' — d<j) d\' — oA ~] 

~ arc 1" |_ h Ja J 

In practice — A<j> may be used for Ji, but if a position computation 
has been made over the line, log h will be immediately available- 
The change in the azimuth a' at B t of the line B x A x for given changes 
in the coordinates of A x and B x may usuaQy be taken the same as 
the change in a , the azimuth al A x of the line A x B x * If the point 
A x is fixed d<j> and dX are zero, and if B x is fixed d(f>' and dX r are zero. 

This formula will now be applied to three examples, first, the 
adjustment of a quadrilateral, next the adjustment of three new 
points connected with a number of fixed points, and, lastly, to a 
figure involving a closure in geographic position. The steps to be 
taken and the precautions to be observed will be explained as they 
arise in the course of the examples. 


* For more exact formula to be used with longer lines, see Dr. F. R. Helmert’s Hohere Geodasie, vol. 1, 
pp. 495 and 496. For such lines some of the approximations made in the derivation here given are no 
longer permissible. 














94 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

In all cases treated by this method, however complicated they may 
be, a start is made with the assumed positions of the points to be 
determined and the assumed azimuths and lengths of the lines sighted 
over. These positions, azimuths, and lengths must be consistent 
with each other and not too far from the final result so that the 
corrections to the assumed quantities are in fact small, as is implied 
in the development of the formulas. Otherwise it is not important 
how these preliminary quantities are found. 

ADJUSTMENT OF A QUADRILATERAL WITH TWO POINTS FIXED 

As a simple example a quadrilateral, A v A 2 , A 3 , A 4 , with two 
points, A t and A 2 , fixed is adjusted. The coordinates of A x and A 2 
and the length and direction of the line A 1 — A 2 are fixed as shown 
in the first lines of the position computation that follows. The angles 
of the preliminary computation of the triangles are obtained from 
the list of directions. To obtain the preliminary positions, directions 
and lengths, the triangles A lf A 2 , A 3 , and A 2 , A 3 , A 4 were made to close 
by correcting each angle by approximately one-third of the error of 
closure as indicated in the triangle computation. This determined the 
entire quadrilateral. In each of the other triangles two sides and 
an included angle became known and thus their remaining parts were 
computed. 

List of observed directions * 

AT As AT A 3 


Station 

\ 

Direction f 

Station 

Direction f 

Initial 

0 00 00.0+Zi 

Initial 

0 00 OO.O +23 

Ai 

0 00 00.0+fli 

At 

0 00 OO.O +07 

As 

101 44 45.1+02 

As 

31 03 42.5+08 

A\ 

133 53 46.3+03 

Ai 

61 47 35.0+09 


AT Ax 


AT Ai 

Initial 

0 00 00.0+22 

Initial 

0 00 00.0+24 

A 3 

0 00 00 . 0+04 

As 

0 00 00.0 +010 

Ai 

26 40 23.5+0 5 

Ax 

25 15 16.2+0,i 

As 

47 31 20.2+06 

As 

116 47 20.0+0,2 


* See fig. 1 on p. 16. 

t Each observed value has its symbolic correction affixed. 













APPLICATION OF LEAST SQUARES TO TRIANGULATION 


Preliminary computation of triangles 


Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 


O 

/ 

n 

n 

n 

tr 

U 


A*-A i 








3.772745 

A 3 

30 

43 

52.5 

+ 0.8 

53.3 



0.291566 

A 2 

101 

44 

45. 1 

+ 0.7 

45.8 

0.1 

45.7 

9.990809 

Ay 

47 

31 

20.2 

+ 0.8 

21.0 



9.867787 





+ 2.3 


0.1 



A s-Ax 








4.055120 

A3-A2 








3.932098 

A2-A1* 








3.772745 

a 4 

25 

15 

16.2 


17.3 



0.369934 

A 2 

133 

53 

46.3 


45.8 

0.1 

45.7 

9.857694 

Ax 

20 

50 

56.7 


57.0 



9.551339 





+ 0.9 


0.1 



Ax-Ax 








4.000373 

A 4- A 2 








3.694018 

A 2- A 3 






0 


3.932098 

A 4 

116 

47 

20.0 

- 1.2 

1S.S 

0.1 

18.7 

0.049306 

A 0 

32 

09 

01.2 

- 1.2 

00.0 



9.726024 

A3 

31 

03 

42.5 

- 1.2 

41.3 



9.712614 





- 3.6 


0.1 



Ax~A 3 








3.707428 

A x~ A 2 








3.694018 

Ax-A 3 * 








4.055120 

Ax 

91 

32 

03.8 


01.5 

0.1 

01.4 

0.000156 

Ax 

26 

40 

23.5 


24.0 



9.652153 

a 3 

61 

47 

35.0 


34.6 



9.945097 





— 2.2 


0.1 



A 4 -A 3 








3.707429 -1 

Ax-A x 








4.000373 


* This triangle is computed from two sides and the included angle. 


91865 °— 15 - 7 































96 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 . 


Preliminary position computation , 


STATION A 3 








0 

/ 

tt 

a 

A 2 to A i 




156 

20 

26.6 

Second angle 

A \and A 3 




+101 

44 

45.8 

a 

A 2 to A 3 




258 

05 

12.4 

Ja 






+ 

8 

05.9 







180 

00 

00 . 00 

a' 

A 3 to A 2 




78 

13 

18.3 





First angle of triangle 

30 

43 

53.3 


O 

/ 

tt 






<f> 

60 

56 

01.089 

A 2 

X 

149 

34 

19. 237 

J(j> 

+ 


56. 720 


JX 

— 

9 

15. 877 

4>’ 

60 

56 

57. 809 

a 3 

X' 

149 

25 

03.360 


1 st term 

2d and 3d 1 
terms / 

— J<f> 


60 56 29 

ft 

-57.0388 
+ 6.3184 


-56. 7204 


s 

COS a 

B 


3.932098 
9.314773 
8.509299 


1. 756170 


s 2 

sin 2 a 
C 


7.86420 
9. 98109 
1.65750 


9.05279 

+0.3183 

+0. 0001 


h 2 

D 


3.512 
2.322 


5. 834 



s 

sin a 

A' 

sec (f>' 

3.932098 
9.990544 
8 . 508600 
0. 313737 

J\ 

sin K <£+<£') 

2. 744979 

9. 941572 



2. 744979 


2. 686551 



tt 


tt 


J\ 

-555. 8774 

—Ja 

-485.90 


STATION A 4 








0 

I 

tt 

a 

A 2 to A 3 




258 

05 

12.4 

Second angle 

A 3 and A 4 




+ 32 

09 

00.0 

a 

A 2 to A i 




290 

14 

12.4 

Ja 






+ 

4 

29.0 







180 

00 

00.00 

a' 

A 4 to A 2 




110 

18 

41.4 





First angle of triangle 

116 

47 

18.8 


0 

t 

ft 






4> 

60 

56 

01.089 

A 2 

X 

149 

34 

19. 237 

j<j> 

— 


55.340 


J\ 


5 

07. 794 

<i>' 

60 

55 

05. 749 

a 4 

X' 

149 

29 

11.443 



O / // 

60 55 33 

tt 

s 

COS a 

B 

3.694018 
9. 53S951 
8 . 509299 

s 2 

sin 2 a 

C 

7.38804 
9.94466 
1. 65750 

h 2 

D 

3. 484 
2.322 

1 st term 

2d and 3d \ 
terms / 

+55. 2418 

+ 0.0979 

h 

1. 742268 


8.99020 

+0. 0978 
+0.0001 


5.806 

-J<t> 

+55.3397 







3. 694018 
9. 972328 
8 . 508601 

J\ 

2. 488260 

0. 313313 

sin §( < £+ , £ , ) 

9. 941507 

2. 488260 


2. 429767 

tt 


tt 

-307. 7939 

—Ja 

-269.0 








































































APPLICATION OF LEAST SQUARES TO TKIANGULATION 


97 


secondary triangulation. 


STATION A 3 


a Alto At 
Third angle A 3 and A 2 


Ja 


J<f> 




1 st term 
2 d,3d,and \ 
4th terms / 


Ai to A 3 


A 3 to A i 


o 

t 

It 

60 

58 

56.416 

— 

1 

58.607 

60 

56 

57.809 


Ai 

A 3 



o 

336 

-47 

I 

18 

31 

II 

08.4 

21.0 


288 

46 

47.4 


+ 

10 

24.3 


180 

00 

00.00 


108 

57 

11.7 

-.1 

X 

149 

36 

57.360 

JX 

— 

11 

54.000 

X' 

149 

25 

03. 360 


O 

/ tt 

60 

57 57 


It 

+ 118.0810 

+ 

0. 5258 

+ 118.60.8 


5 

COS a 

B 

4.055120 
9. 507765 
8. 509295 

52 

sill 2 a 
C 

8.11024 
9.95248 

1. 65837 

h 2 

D 

4.144 
2. 322 

-h 

s 2 sin 2 « 
E 

h 

2.072180 


9. 72109 


6. 466 





+0. 5261 
+0.0003 
-0.0006 





2.072 
8.063 
6.640 


6. 775 


s 

sin a 
A' 

sec <f>' 


JX 


4.055120 
9.976241 
8.508600 
0. 313737 


2. 853698 

it 

-713.9996 


JX 

sin £(<£+<£') 


—Ja 


2. 853698 
9.941676 


2. 795374 


-624.27 


STATION Ai 







0 

I 

It 

a 

A 3 to A 2 



78 

13 

18.3 

Third angle 

A 4 and A 2 



- 31 

03 

41.3 

a 

A 3 to A 4 



47 

09 

37.0 

Ja 





— 

3 

36.8 






180 

00 

00.00 

a' 

A 4 to A 3 



227 

06 

00.2 


O 

/ 

It 






<{> 

60 

56 

57. 809 

A 3 

X 

149 

25 

03.360 

Jcf> 

— 

1 

52.060 


JX 

+ 

4 

08.084 


60 

. 55 

05. 749 

A 4 

X' 

149 

29 

11.444 



O t ft 

60 56 02 

It 

s 

COS a 

B 

3. 707428 
9. 832477 
8 . 509298 

s 2 

sin 2 a 

C 

7. 41486 
9. 73052 
1.65808 

h 2 

D 

1 st term 

2d and 3d \ 
terms / 

+ 111.9962 
+ 0.0639 

h 

2.049203 


8.80346 

+0.0636 
+0.0003 


-J<t> 

+112.0601 






s 

sin a 

A' 

sec <t>’ 

3. 707428 
9.865257 
8 . 508601 
0.313313 

JX 

sin J(<£+<£') 

2.394599 
9.941540 


2. 394599 


2.336139 


It 


It 

JX 

+248.0841 

—Ja 

+216.8 

































































98 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


FORMATION OF OBSERVATION EQUATIONS 

The observation equations used in making the adjustments are 
formed on the assumptions of the direction method.* Each point¬ 
ing of the telescope is treated as an independent observation and the 
sum of the squares of the corrections to the separate pointings is to 
be made a minimum. A single pointing, however, taken by itself 
determines nothing, for if each of the pointings at a station be changed 
by the same amount the set of pointings has the same significance as 
before. The effect is simply a change in the zero direction, which is 
a purely arbitrary matter. If a set of corrections to directions at a 
point has been determined by any method and the mean of these 
corrections is not zero, the sum of the squares of these corrections 
can always be diminished by subtracting from each correction the 
mean of all of the corrections so that the algebraic sum of the reduced 
corrections is zero. Hence in any set of directions adjusted by the 
method of least squares the algebraic sum of the corrections at a 
point is zero.f To allow for this change of zero direction, or for the 
constant correction to all directions at a point, an unknown constant 
correction, “ 2 ,” is introduced into all equations expressing the 
results of observations at a point, a different “ 2 ” for each point 
where observations are taken. 

The observation equation may be written, 

Assumed azimuth + da — observed azimuth + 2 — v = 0. 

The coefficients of the dfis and £Ts come from the last equation on 
page 93. As a sample, take those in the expression for t? 9 . A x cor¬ 
responds to the A x and A 3 to the B x of the explanation of the for¬ 
mulas. Sin or, cos or, h, and AX come from the position computation 
on page 97. 


log sin a 9. 9762n 

4. 7984n 

4. 7984n 

log cos a 9. 5078 

log h 2. 0722 

log A\ 2. 8537n 

colog arc 5. 3144 

2. 7262n 

1. 9447 

4. 7984n 

Number —532 

Number +88 


The observed angles in the following formation of equations come 
from the list of directions on page 94. 

Azimuth A 2 to A x (initial direction). 156 20 26.6 

Observed angle initial direction to A x . 0 00 00. 0— 

Observed azimuth A 2 to A x . 156 20 26. Q—z l +v l 

Assumed azimuth A 2 to A t . 156 20 26. 6+c?a 

Assumed azimuth—observed azimuth. 0=0. Q-\-da-\-z x —v x 

* See Wright and Hayford, Adjustment of Observations, Chap. VII. 

t This does not necessarily hold good when a line whose direction has already been fixed enters into 
the set. 











APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


99 


Azimuth A 2 to A x is fixed. Therefore doc = 0 and v x = z x 

Azimuth A 2 to A x (initial direction). 156 20 26.6 

Observed angle initial direction to A 3 . 101 44 45.1— z x -\-v 2 

Observed azimuth A 2 to A z . 258 05 11. 7 — z x -\-v 2 

Assumed azimuth A 2 to A 3 . 258 05 12. A-\-da 

Assumed azimuth—observed azimuth. 0=+0. 7-\-da-\-z x —v 2 

da=- 7S0d(f> 3 - 75dX 3 . Therefore v 2 = z x - 730 d(f> 3 - 75dX 3 + 0.7. 

Azimuth A 2 to A x (initial direction). 156 20 26. 6 

Observed angle initial direction to A 4 . 133 53 46. 3— z x -{-v 3 • 

Observed azimuth A 2 to A 4 . 290 14 12. 9— z x -\-v 3 

Assumed azimuth A 2 to A 4 . 290 14 12. 4-j-da 

Assumed azimuth—observed azimuth. 0= —0. 5+</or+z 1 — v 3 

da=- 1212 dcj) 4 + 218 dX 4 . Therefore v 3 = z x - 1212 <£0 4 + 218 £A 4 -0.5 

In the same way at A x v 4 =z 2 —532 £0 3 +88 £+ —0. 8 
v 5 =z 2 -447 £0 4 +221 £4-0. 3 
v 6 =z 2 

Azimuth A 3 to A 4 (initial direction). 47 09 37. 0 

Observed angle initial direction to A 4 . 0 00 00. 0 -z 3 -j-v 7 

Observed azimuth A 3 to A 4 . 47 09 37. 0— z 3 -\-v 7 

Assumed azimuth A 3 to A 4 . 47 09 37. 0-j-da 

Assumed azimuth—observed azimuth. 0=0. 0+efo+z 3 — v 7 

da= +918 (dcf) 4 — d(j> 3 ) +414 (^ 4 —^ 3 ). Therefore v 7 = z 3 — 918 d(j> 3 
-414 £4 + 918 £<^ 4 + 414 £4 + 0.0 
Similarly 

v 8 = 23-730 £0 3 -75 £4-1. 2 
v 9 =^-532 £<£ 3 +88 £+-0. 4 

v l0 =z 4 -1211 £0 4 +218 £4+0. 0 

Vu =2 4 -447 £0 4 +221 £4+1.1 

v x2 =z 4 -918 £0 3 -414 £4+918 £0 4 +414 £4-1. 2 

We have then the set of observation equations: 

v x =z x 

v 2 =z x —730 £0 3 —75 £4+0. 7 
v 3 =z x -1212 £04+218 £4-0. 5 

v 4 = 2 2 —532 £03+88 £4-0. 8 
v 5 = 2 2 —447 £04+221 £4-0.3 

~Z 2 

v 7 = 2 3 —918 £03-414 £4+918 £0 4 +414 £4+0.0 
v 8 =2 3 —730 £03-75 £4-1. 2 
v 9 = 2 3 —532 £03+88 £4-0. 4 

v xo =z 4 -1211 £0 4 +218 £4+0. 0 

v xx ==z 4 —447 £0 4 +221£ 4+1* 1 

-y 12 =2 4 —918 £03-414 £4+918 £0 4 +414 £4-1. 2 

These equations contain z’s which are of no particular interest in 
themselves. The normal equations might be formed and the z’s 
eliminated in the regular way, but this work is made easier by the 
following mechanical rule, the effect of which is to form at once the 

















100 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


reduced normal equations with the z’s eliminated. For the proof 
of the rule and further particulars see Jordan’s Handbuch der Ver- 
messungskunde, Yol. I, pages 151-171, of the third edition. Each 
direction is assumed to have equal weight. Write the observation 
equations dropping the z’s and giving each unit weight. Add 
together as they stand all observation equations containing the z 
for any particular point. Drop the z term out and treat the result¬ 
ing equation as a new observation equation with a negative weight 
equal to — l/r, where r is the total number of directions, both fixed 
and to be determined, that have been observed at the point in ques¬ 
tion. To reduce the new fictitious observation to unit weight it 

must be multiplied through by 

Table 1 below shows the coefficients of the unknowns, the coeffi¬ 
cients formed by adding the equations containing any partic¬ 
ular z, and the weights. Table 2 shows these equations divided 
through by 100 for convenience. This has no effect on the relative 
weights. The table also contains the fictitious observation equa- 


-y = ^-\jr i where i = V — 1. 


tions obtained by multiplying the sum equation 



From 


Table 2 the normal equations which do not contain the z’s are 
formed in the ordinary way for observation equations of equal 
weight, using the i’s strictly according to algebraic laws. Thus in 
the first line of Table 2 ( —4.23i) 2 contributes to the first diagonal 
coefficient not +17.8929 but + 17.8929i 2 , or —17.8929, and ( —4.23i) 
X + (1.26i) contributes toward the side coefficient not —5.3298, but 
— 5.3298i 2 or +5.3298. 


Table for formation of normals , No. 1 




d<f> 3 

d\ 3 

S<f> 4 

S \4 

l 

V 

VF 


2l 









1 






1 

i 


1 

- 730 

- 75 



+0.7 

1 

i 


1 



-1212 

+218 

—0.5 

1 

i 

* Sum 

3 

- 730 

- 75 

-1212 

+218 

+0.2 

-i 

0.58 i 


Zj 









1 

- 632 

+ 88 



-0.8 

1 

1 


1 



- 447 

+221 

-0.3 

1 

1 


1 






1 

1 

Sum 

3 

- 532 

+ 88 

- 447 

+221 

-1.1 

— i 

0.58i 


z 3 









1 

- 918 

-414 

+ 918 

+414 

+0.0 

1 

1 


1 

- 730 

- 75 



-1.2 

1 

1 


1 

- 532 

+ 88 



-0.4 

1 

1 

Sum 

3 

-2180 

-401 

+ 918 

+414 

-1.6 

-i 

0.58 i 


24 









1 



-1211 

+218 

+0.0 

1 

1 


1 



- 447 

+221 

+1.1 

1 

1 


1 

- 918 

-414 

+ 918 

+414 

-1.2 

1 

1 

Sum 

3 

- 918 

-414 

- 740 

+853 

-0.1 

-h 

0.58i 












APPLICATION OF LEAST SQUARES TO TRIANGULATION. 101 


Table for formation of normals, No. 2 



d<t>3 

5 X 3 

Hi 

d\i 

l 

2 

2 

- 7.30 

-0.75 



+0.007 

- 8.043 

3 



-12.12 

+2.18 

-0.005 

- 9.945 

Zi 

- 4.231 

-0.44! 

- 7.031 

+1.261 

+0.001161 

-10.438841 

4 

- 5.32 

+0.88 



-0.008 

- 4.448 

5 



- 4.47 

+2.21 

-0.003 

- 2.263 

Zi 

- 3.09! 

+0.511 

- 2.591 

+1.281 

-0.006381 

- 3.89638! 

7 

- 9.18 

-4.14 

+ 9.18 

+4.14 

+0.0 

0.0 

8 

- 7.30 

-0.75 



-0.012 

- 8.062 

9 

- 5.32 

+0.88 



-0.004 

- 4.441 

Zi 

-12.641 

-2.331 

+ 5.321 

+2.401 

-0.009281 

- 7.259281 

10 



- 12 . ir 

+2.18 

+0.0 

- 9.93 

11 



- 4.47 

+2.21 

+0.011 

- 2.249 

12 

- 9.18 

-4.14 

+ 9.18 

+ 4.14 

-0.012 

- 0.012 

Zi 

- 5.32! 

-2. 401 

- 4.291 

+4.951 

-0.000581 

- 7.060581 


Normal equations 



d<{>3 

8 X 3 

Hi 

dXi 

ij 

2 

1 

+116. 2166 

+35.0927 

-161.8628 

-10.0554 

+0.0753 

- 20.5336 

2 


+25.3104 

- 75.6831 

-16.9056 

+0.0236 

- 32.1620 

3 



+399. 2176 

+24.0721 

-0.0468 

+ 185.6970 

4 




+20.0637 

-0.01105 

+ 17.16375 


The forward and back solution of the normals, conducted accord¬ 
ing to the Doolittle method, is next shown. 

To compute the v’s from the observation equations a knowledge 
of the z' s is required. Substitute the d</>’ s and dA’s in the right-hand 
side of the sum equation formed from the observation equations 
that contain the z in question as if the z were not there and divide 
the result of the substitution by the weight —r. As a check the 
sum of the v’s about a point should equal zero. The computation of 
the v’s is shown in the table on page 102. Below each v as computed 
to 3 decimals is given its value as adopted and reduced to 1 decimal. 

Following the computation of the v’s there is given a computation 
of the triangles using the adjusted directions. 


Solution of normals 


8<f>3 


Hi 

DXi 

i} 

2 

+116. 2166 

+35.0927 

-161.8628 

-10.0544 

+0.0753 

- 20.5336 

Hi 

- 0.301959 

+ 1.392768 

+ 0.086523 

-0.000648 

+ 0.176684 


+25.3104 

- 75.6831 

-16.9056 

+0.0236 

- 32.1620 

1 

-10.5966 

+ 48.8759 

+ 3.0363 

-0.0227 

+ 6.2003 


+14.7138 

- 26.8072 

-13.8693 

+0.0009 

- 25.9618 


dk 

+ 1.821909 

+ 0.942605 

-0.000061 

+ 1.764452 



+399. 2176 

+24.0721 

-0. 0468 

+185.6970 


1 

-225. 4373 

-14. 0048 

+0.1049 

- 28.5985 


2 

- 48.8403 

-25. 2686 

+0.0016 

- 47.3000 



+124.9400 

-15.2013 

+0.0597 

+109. 7984 



84> i 

+ 0.121669 

-0. 000478 

- 0.878809 




+20. 0637 

-0. 01105 

+ 17.16375 



1 

- 0. 8700 

+0.00652 

- 1.77663 



2 

-13.0733 

+0. 00085 

- 24.47172 



3 

- 1.8495 

+0.00726 

+ 13.36904 




+ 4.2709. 

+0.00358 

+ 4.27448 




SXi 

-0.000838 

- 1.000838 































V 

102 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Back solution 


dXi 

<50 4 

<5/3 

<503 

-0.00084 

-0.00048 
-0.00010 

-0. 00006 
-0.00079 
-0.00106 

-0.00065 
-0. 00007 
-0. 00081 
+0. 00058 

-0.00084 

-0.00058 

-0. 00191 

-0.00095 


Computation of corrections 


l=Zl 

2 

3 

Zl 

-0.519 
-0.5 

+0. 6935 
+0.1432 
+0.7 
-0. 519 

+0. 7030 
-0.1831 
-0.5 
-0. 519 

+0. 6935 
+0.1432 
+0. 7030 
-0.1831 
+0.2 

+ 1.018 
+ 1.0 

-0. 499 
-0.5 

+ 1. 5566-;—3 
-0. 519 

4 

5 

6= 22 

22 

+0. 5054 
-0.1681 
-0.8 
+0. 230 

+0. 2593 
-0.1856 
-0.3 
+0. 230 

+0. 230 
+0.3 

+0. 5054 
-0.1681 
+0. 2593 
-0.1856 
-1.1 

-0. 233 
-0.2 

+0. 004 

0.0 

-0.689 3 

+0. 230 

7 

8 

9 

23 

+0. 8721 
+0. 7907 
-0. 5324 
-0. 3478 
0.119 

+0.6935 
+0.1432 
-1.2 
-0.119 

+0. 5054 
-0.1681 
-0.4 
-0.119 

+2.0710 
+0. 7659 
-0. 5324 
-0. 3478 
-1.6 

-0. 482 
-0.5 

-0.182 
-0.2 

+0.664 
+0.7 

+0.3567-=--3 
-0.119 

10 

11 

12 

24 

+0. 7024 
-0.1831 
-0. 425 

+0. 2593 
-0.1856 
+1.1 
-0. 425 

+0. 8721 
+0. 7907 
-0. 5324 
-0. 3478 
-1.2 
-0. 425 

+0. 8721 
+0. 7907 
+0. 4292 
-0. 7165 
-0.1 

+0.094 
+0.1 

+0. 749 
+0.7 

+ 1.2755-=--3 
-0. 425 

-0. 842 
-0.8 


> 


. ) 














































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 103 


Adjusted computation of triangles 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 



O 

/ 

„ 

„ 

„ 

n 

// 



At-Ai 








3.772745 

-8+9 

Az 

30 

43 

52.5 

+0.3 

52.8 

0.0 

52.8 

0.291568 

-1+2 

A 2 

101 

44 

45.1 

+ 1.5 

46.6 

0.1 

46.5 

9.990809 

-4+6 

A\ 

47 

31 

20.2 

+0.5 

20.7 

0.0 

, 20.7 

9.867787 






+2.3 


0.1 




A3-A1 








4.055122 


A3-A2 








3.932100 


A2-A1 








3.772745 

-10+11 

Ai 

25 

15 

16.2 

+0.6 

16.8 

0.0 

16.8 

0.369936 

-1+3 

A 2 

133 

53 

46.3 

0.0 

46.3 

0.1 

46.2 

9.857693 

-5+6 

A\ 

20 

50 

56.7 

+0.3 

57.0 

0.0 

57.0 

9.551339 






+0.9 


0.1 




A i-A 1 








4.000374 


A i-A 2 








3.694020 


A2-A3 








3.932100 

-10+12 

A< 

116 

47 

20.0 

-0.9 

19.1 

0.1 

19.0 

0.049306 

-2+3 

A 2 

32 

08 

61.2 

-1.5 

59.7 

0.0 

59.7 

9. 726023 

-7+8 

Az 

31 

03 

42.5 

-1.2 

41.3 

0.0 

41.3 

9. 712614 






-3.6 


0.1 




At-Az 








3. 707429 


At-A 2 








3.694020 


A\-Az 








4.055122 

-11+12 

At 

91 

32 

03.8 

-1.5 

02.3 

0.1 

02.2 

0.000156 

-4+5 

A 1 

26 

40 

23.5 

+0.2 

2.3.7 

0.0 

23.7 

9.652151 

-7+9 

Az 

61 

47 

35.0 

-0.9 

34.1 

0.0 

34.1 

9.94.5096 






-2.2 


0.1 




At-Az 








3. 707429 


At~Ai 








4.000374 


ADJUSTMENT OF THREE NEW POINTS CONNECTED WITH SEVERAL 
FIXED POINTS BY VARIATION OF GEOGRAPHIC COORDINATES 

GENERAL STATEMENT 

The method of adjustment by geographic coordinates seems to be 
especially suitable for the adjustment of a few new points depending 
upon a number of fixed points. The number of normal equations 
in such case is 2 n, n being the number of new points. In the figure 
used the number of condition equations would be 15, which would 
form a very intricate set of normals. By the method of coordinates 
the number of normal equations is only six. 

The adjustment of figure 6 is carried out in two different ways, 
the first one being more rigorous but a trifle longer than the second. 
The first method corresponds in its treatment of observed directions 
to the method developed in Jordan’s Vermessungskunde, volume 1, 
pages 144-173, of the third edition. The second method resembles 
somewhat the method given by Jordan on pages 173-179 for the 
approximate treatment of the z 's and corresponds in its treatment of 
fixed directions to the ordinary practice of the Coast and Geodetic 
Survey for subsidiary triangulation as treated by the method of 
condition equations. 





















104 COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 28. 




APPLICATION OF LEAST SQUARES TO TRIANGULATION. 105 

The solution by the method of condition equations was carried 
out for figure 6 and gave almost the same results, the greatest dif¬ 
ference in the correction to a direction being 0.2". This difference 
was quite to be expected in view of the different formulas and the 
fact that the fixed positions, distances, and azimuths may not be 
strictly consistent with each other to the last figure given. 

FIRST METHOD 

The first method is fundamentally the same as the method used in 
the adjustment of the quadrilateral previously given, but the greater 
complication of the figure, particularly the great number of fixed 
lines, brings to light points that need mention. The groundwork »of 
the adjustment by either method is shown in the tables of observed 
directions and of fixed positions, azimuths, and lengths which follow. 
In the list of directions the names of stations that are sighted on 
over fixed lines are shown in heavy type. For these stations the 
directions corrected from a previous adjustment are also shown. In 
forming the table of triangles for the preliminary computation these 
corrected directions were taken with the directly observed directions 
of new points in order to obtain such of the angles in the column 
“Observed angle” as have a fixed line for one of its sides.* No 
particular procedure to obtain the consistent set of positions, azi¬ 
muths, and lengths necessary to form the observation equations is 
essential to the method. In this particular case the corrections to 
the angles of the triangles Gunner-Larrabee-Mam, Cranberry Point- 
Gunner-Lubec Channel Lighthonse, and Telegraph-Cranberry Point- 
Gunner were arbitrarily assumed as shown in the table of preliminary 
computation of triangles. These assumptions, with the lines already 
fixed, determined enough parts in every one of the other triangles to 
make possible its solution with results as shown in the table. 

Following the table of triangles the necessary preliminary compu¬ 
tation of positions is included. 


* This corresponds to the idea followed out in the second method of solution, but in the preliminary 
computation this is of no consequence, as is shown in the next sentence. 



106 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Lists of directions 

AT INDIAN POINT 


AT LARRABEE 


Stations observed 

Observed 

directions 

Seconds 

after 

adjust¬ 

ment* 


Of ff 

// 

Indian Point 

0 00 00.0 

56.7 

Mam 

281 47 48.4 

47.9 

Lubec Channel Light¬ 
house 

312 58 58.5 

55.0 

Lubec Church Spire 

315 12 19.0 

24.9 

Gunner 

325 58 24.9 


Duck 

33G 12 51.0 

50.6 


AT DUCK 


Stations observed 

Observed 

directions 

Seconds 

after 

adjust¬ 

ment* 


Of ff 

ff 

Larrabee 

0 00 00.0 

01.5 

Mam 

6G 55 01.0 

01.2 

Lubec Channel Light¬ 
house 

95 24 38.4 

41.2 

Gunner 

114 00 37.5 


Lubec Church Spire 

117 33 55.5 

71.9 


AT MAM 


Indian Point 

0 00 00.0 

55.4 

Larrabee 

34 52 43. G 

46.7 

Cranberry Point 

317 37 23.5 


Lubec Channel Light¬ 
house 

Gunner 

318 56 54.1 

55.2 

322 01 44.8 


Duck 

336 18 52.1 

52.5 


Lubec Channel Light- 

0 00 00.0 

59.9 

house 



Gunner 

33 11 41.8 


Larrabee 

269 09 40.7 

40.9 

Mam (computed) 

336 10 

44.0 


AT GUNNER 


AT CRANBERRY POINT 


Indian Point 

0 

00 

00.0 

Larrabee 

31 

57 

41.4 

Mam 

94 

56 

05.9 

Lubec Channel Light- 

101 

48 

54.9 

house 




Telegraph 

Lubec Church Spire 

168 

186 

28 

30 

59.0 

29.8 

Cranberry Point 

191 

54 

53.6 

Duck 

346 

14 

12.0 


AT TREAT 2 


Cranberry Point 

Lubec Church Spire 

0 00 00.0 
17 34 38.1 

45.4 


Gunner 

Lubec Channel Light¬ 
house 
Mam 
Telegraph 
Lubec Church Spire 
Treat 2 


0 

00 

00.0 

75 

45 

12.8 

78 

37 

05.9 

1.53 

39 

19.7 

174 

08 

32.3 

191 

52 

35.2 


AT TELEGRAPH 


Cranberry Point 

0 00 00.0 


Gunner 

2 54 53.2 


Lubec Channel Light- 

29 52 44.5 


house 



Lubec Church Spire 

231 21 00.1 



* Refers to final values of heavy lines in Fig. G, p. 104, obtained from a previous adjustment. 


List 0 f fixed positions 


Station 

Latitude and 
longitude 

Azimuth 

Back azimuth 

Xo station 

Loga¬ 
rithm 
of dis¬ 
tance 


O 

r 

ft 

O 

r 

ft 

O 

/ 

ft 



Lubec Church Spire 

44 

51 

38. 470 










66 

59 

17.418 









Lubec Channel Light- 

44 

50 

31. 652 









house 

66 

58 

38.299 









Treat 2 

44 

52 

44. 333 

354 

45 

17.5 

174 

45 

23.5 

Lubec Church Spire 

3.309982 


66 

59 

25.919 









Indian Point 

44 

50 

03. 537 

114 

48 

33.8 

294 

47 

33.5 

Lubec Channel Light- 

3.315762 


66 

57 

12. 788 







house 






136 

58 

04.6 

316 

56 

36.7 

Lubec Church Spire 

3. 603123 

Larrabee 

44 

49 

10.841 

152 

22 

33.8 

332 

21 

51.9 

Lub'ec Channel Light- 

3. 449573 


66 

57 

38. 857 







house 






154 

36 

03.7 

334 

34 

54.3 

Lubec Church Spire 

3. 702872 





199 

23 

35.7 

19 

23 

54.1 

Indian Point 

3.236668 

Duck 

44 

50 

33.886 

355 

36 

23.1 

175 

36 

29.4 

Larrabee 

3.410111 


66 

57 

47. 822 

86 

26 

42.1 

266 

26 

06.5 

Lubec Channel Light- 

3.045619 











house 


Mam 

44 

49 

57.369 

225 

14 

19.0 

45 

14 

53.3 

Lubec Channel Light- 

3.176968 


66 

59 

26. 892 







house 


\ 




242 

36 

16.3 

62 

37 

26.2 

Duck 

3.389282 





266 

17 

19.2 

86 

18 

53. 8 

Indian Point 

3. 470097 





301 

10 

10.5 

121 

11 

26.7 

Larrabee 

3. 443126 

















































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 107 


Preliminary computation of triangles 


Symbol 

Station 

Observed 

anglo 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

Spheri 

cal 

excess 

Plane angle- 

Loga¬ 

rithm 



o 

/ 

// 

// 



O / // 


-1+ 3 

Duck-Larrabee 








3 410111 

Gunner 

45 

43 

29.4 

+ 4.8 

34.2 



0.145080 

+24 

Duck 

124 

01 

60.9 

— 1.9 

59.0 



9.918405 

—22 

Larrabee 

10 

14 

25.7 

+ 1.1- 

26.8 



9.249896 


Gun ner-1 >arabee 
Gunner-Duck 




+ 4.0 












3. 473596 

2.805087 

-1+ 4 

Duck-Mam 








3.389282 

Gunner 

10S 

41 

53.9 

+ 3.6 

57.5 



0.023552 

+24 

Duck 

57 

00 

57. 8 

- 1.9 

55.9 



9.923668 

—21 

Mam 

14 

17 

07.7 

- 1.1 

06.6 



9.392254 


Gunner-Mam 

Gunner-Duck 




+ 0.6 












3.336502 

2. 805088-1 


Duck-Lubec Chan- 








3.045610 


nel Lighthouse 








-1+ 5 

Gunner 

115 

34 

42.9 

-1- 7. 1 

50.0 



0. 0H 803 

+24 

Duck 

Lubec Channel 

33 

11 

41.9 

35.2 

- 1.9 

- 5.2 

40.0 

30.0 


31 13 30.0 

9. 738370 

9. 714665 


Lighthouse 
Gunne r-L u b e c 








2. 828792+1 


Channel Light¬ 
house 










Gunner-Duck 








2 80.5087 


Indian Point-Lar- 








3.236668 


rabee 








-2+ 3 

Gunner 

31 

57 

41.4 

+ 7.0 

48.4 



0. 276234 

+23 

Indian Point 

114 

00 

36.0 

+ 2.5 

38.5 



9.960694 

-22 

Larrabee 

34 

01 

32.0 

+ 1.1 

33.1 



9. 747852 


Gunner-Larrabee 




+ 10.6 












3. 473596 


Gunner-Indian 








3.260754 


Point 










Indian Point-Mam 








3.470097 

-2+ 4 

Gunner 

94 

56 

05.9 

+ 5.8 

11.7 



0.001614 

+23 

Indian Point 

47 

05 

36.3 

f 2.5 

38.8 



9. 864792 

-21 

Mam 

37 

58 

10.6 

- 1.1 

09.5 



9.789044 






+ 7.2 






Gunner-Mam 







3. 336503-1 


Gunner-Indian 








3.260755-1 


Point 










Indian Point-Lu- 








3.315762 


hoc Channel 
Lighthouse 









-2+ 5 

Gunner 

101 

48 

54.9 

+ 9.3 

64.2 



0. 009304 

+23 

Indian Point 

18 

35 

56.3 

+ 2.5 

58.8 



9.503728 


Lubec Channel 



68.8 


57.0 


59 34 57.0 

9.935688 


Lighthouse 
Gunne r-L u b e c 








2.828794-1 


Channel Light¬ 
house 










Gunner-Indian 








3.260754 


Point 










Larrabee-Mam 








3.443126 

-3+ 4 

Gunner 

62 

58 

24.5 

- 1.2 

23.3 



0.050223 

+22 

Larrabee 

44 

10 

37.0 

- 1.1 

35.9 



9.843153 

-21 

Mam 

72 

51 

01.9 

- 1.1 

00.8 



9.980247 


Gunner-Mam 




- 3.4 




3.336502 


Gunner-Larrabee 








3.473596 


























108 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

Spheri¬ 

cal 

excess 

Plane angle 

Loga¬ 

rithm 



O 

/ 

// 

// 

// 


O 

/ 

/f 



Larra b e o-L ubec 










3. 449573 


Channel Light¬ 
house 











-3+ 5 

Gunner 

09 

51 

13.5 

+ 2.3 

15.8 





0.027417 

+22 

Larrabee 

12 

59 

29.9 

- 1.1 

28.8 





9.351803 


Lubec Channel 



16.6 


15.4 


97 

09 

15.4 

9.996606 


Lighthouse 

G u n n e r-L ubec 










2.828793 


Channel Light¬ 
house 












Gunner-Larrabee 










3.473596 


L arr abee-Lubec 










3.702872 


church spire 











-3+7 

Gunner 

154 

32 

48.4 

+20.7 

69. 1 



33 

09. 1 

0. 366851 

+22 

Larrabee 

10 

45 

60.0 

- 1.1 

58.9 





9.271388 


Lubec church spire 



71.6 


52.0 


14 

40 

52.0 

9.403873 


Gunner -Lub ec 










3.341111 


church spire 
Gunner-Larrabee 










3. 473596 


Mam-Lubec Chan- 










3.176968 


nel Lighthouse 











-4+5 

Gunner 

6 

52 

49.0 

+ 3.5 

52. 5 





0.921500 

+21 

Mam 

3 

04 

49.6 

+ 1.1 

50.7 





8.730324 


Lubec Channel 



21.4 

16.8 


170 

02 

16.8 

9. 238033 


Lighthouse 

Gunner-Lubec 






2.828792+' 






Channel Light¬ 
house 












Gunner-Mam 










3.336501+1 


Lubec church 










3.603122 


spire-Indian Point 











+ 2-7 

Gunner 

173 

29 

30.2 

-27.7 

02.5 





0.945080 


Lubec church 


56 

55.4 


85.6 


2 

57 

25.6 

8. 712552 

-23 

spire 

Indian Point 

3 

33 

34.4 

- 2.5 

31.9 





8. 792909 


Gunner-In dian 
Point 

Gunner-Lubec 










3.260754 

3.341111 


church spire 











Gunner-Lubec 










2.828793 


Channel Light¬ 
house 











- 9+10 

Cranberry Point 

75 

45 

12.8 

0.0 

12.8 





0.013566 

-5+8 

Gunner 

90 

05 

58.7 

0.0 

58.7 





9.999999 


Lubec Channel 



48.5 


48.5 


14 

08 

48.5 

9.388114 


Lighthouse 
Cranberry Point- 







2.842358 


Lubec Channel 
Lighthouse 












Cranberry Point- 










2.230473 


Gunner 











Gunner-Mam 










3.336502 

- 9+11 

Cranberry Point 

78 

36 

65.9 

-13.1 

52.8 





0.008631 

-4+8 

Gunner 

96 

58 

47.7 

+ 3.5 

51.2 





9.996768 

-20+21 

Mam 

4 

24 

21.3 

— 5.3 

16.0 



r 


8.885341 


Cranberry Point- 
Mam 

Cranberry Point- 




-14.9 






3.341901 

2.230474-' 


Gunner 











Gunner-Lubec 










3.341111 


church spire 










- 9+13 

Cranberry Point 

174 

08 

32.3 

+ 11.5 

43.8 





0.991389 

-7+8 

Gunner 

5 

24 

23.8 

-18.4 

05.4 





8.973748 


Lubec church spire 
Cranberry Point- 



03.9 


10.8 


0 

27 

10.8 

7.897971 

3.306248 


Lubec church 
spire 












Cranberry Point- 










2.230471+2 


Gunner 
































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 109 


Preliminary computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

-10+11 

-20 

L u b e c Channel 
Lighthouse-Mam 
Cranberry Point 

L u b e c Channel 
Lighthouse 

Mam 

Cranberry Point- 
Mam 

Cranberry Point- 
Lubec Channel 
Lighthouse 

O / // 

2 51 53.1 
35.2 

1 19 31.7 

// 

-13.1 

- 6.4 

// 

40.0 

54.7 

25.3 

-13+14 

-19 

Lubec church 
spire-Treat 2 
Cranberry Point 
Lubec church spire 
Treats 

Cranberry Point- 
Treat 2 

Cranberry Point- 
Lube'c church 
spire 

17 43 62.9 
19.0 

17 34 38.1 

- 4.7 

- 4.3 

58.2 

28.0 

33.8 

-15+16 

-12+13 

Lubec church 
spire-Cranberry 
Point 

Telegraph 

Lubec church spire 
Cranberry Point 

T e 1 e g r aph-Cran- 
berry Point 
Telegraph-L u b e c 
church spire 

128 38 59.9 
51 47.5 
20 29 12.6 

-52.6 

+ 14.2 

07.3 

85.9 

26.8 

-15+17 

-6+7 

Lubec church 
spire-Gunner 
Telegraph 

Lubec church spire 
Gunner 

Telegraph-G unner 
Telegraph - Lubec 
church spire 

131 33 53.1 
24 36.1 
IS 01 30.8 

-54.8 

+ 15.8 

- 58.3 
75.1 
46.6 

-16+17 
- 9+12 
-6+8 

Cranberry Point- 
Gunner 

Telegraph 

Cranberry Point 
Gunner 

2 54 53.2 
153 39 19.7 
23 25 54.6 

- 2.2 

- 2.7 

- 2.6 

51.0 

17.0 

52.0 


Telegraph-Gunner 

T e 1 e g r aph-Cran- 
berry Point 


- 7.5 


—10+18 
-10+12 

Cranberry Point- 
Lubec* Channel 
Lighthouse 
Telegraph 

Cranberry Point 
Lubec ” Channel 
Lighthouse 
Telegraph-L ubec 
Channel Light¬ 
house 

T e 1 e g r aph-Cran- 
berry Point 

29 52 44.5 
77 54 06.9 
08.6 

- 9.6 

- 2.7 

34.9 
04.2 

20.9 

-17+18 

-5+6 

Gunner-Lubec 
Channel Light¬ 
house 

Telegraph 

Gunner 

Lubec Channel 
Lighthouse 
Telegraph-L ubec 
Channel Light¬ 
house 

Telegraph-Gunner 

26 57 51.3 
66 40 04.1 
04.6 

- 7.4 
+ 2.6 

43.9 

06.7 

09.4 


Spheri¬ 

cal 

excess 


175 48 54.7 


Plane angle 


Loga¬ 

rithm 


144 41 28.0 


30 52 25.9 


32 58.3 
30 25 15.1 


72 13 20.9 


80 22 09.4 


3.176968 

1.301768 
8.8G3166 

8.303626 

3.341902-1 

2.842362“ 4 


3.309982 

0.516300 
9.761916 
9.479966 
3.588198 

3.306248 


3.306248 


0.107274 
9.710244 
9.544138 
3.123766 

2.957660 


3.341111 

0.125876 
9.704449 
9.490673 
3.171436 
2.957660 


2.230473 

1.293796 

9.647167 

9.599497 


3.171436 
3.123766 


2.842358 

0.302657 
9.990244 
9.978751 

3.135259+1 
3.123766 
2.828793 


0.343516 
9.962951 
9.999127 

3.135260 


3.171436 




























110 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation , 

STATION GUNNER 








O 

/ 

n 

ct 

Duck to Lubec Channel Lighthouse 


86 

26 

42.1 

Second angle 

Lubec Channel Lighthouse and Gunner 


+ 33 

11 

40.0 

a 

Duck to Gunner 



119 

38 

22.1 

Ja 






— 


17.8 







180 

00 

00.00 

a’ 

Gunner to Duck 



299 

38 

04.3 





First angle of triangle 

115 

34 

50.0 

(f> 

O 

44 

/ 

50 

n 

33.886 

Duck 

X 

66 

57 

47. 822 

J<j> 

+ 


10. 227 


JX 

+ 


25.265 

<t>' 

44 

50 

44.113 

Gunner 

X' 

66 

58 

13.087 



Off/ 

s 

2. 805087 

s 2 

5.6102 


44 50 39 

COS a 

9. 694202 

sin 2 a 

9.8782 

// 

B 

8. 510480 

C 

1. 4016 

1st term 

-10.2275 

h 

1.009769 


6.8900 

2d term 

+ 0.0008 





-A4> 

-10.2267 






s 

sin a 
A' 

sec <f>' 


JX 


2.805087 
9.939097 
8. 508994 

JX 

0.149348 

sin 

1. 402526 


/ 1 


+25.2654 

—Ja 


1.402526 
9.848301 


1. 250827 


+ 17. 82 


STATION CRANBERRY POINT 








O 

/ 

// 

a 

Gunner to Lubec Channel Lighthouse 


55 

12 

54.3 

Second angle 

Lubec Channel Lighthouse and Cranberry Point 

+ 90 

05 

58.7 

a 

Gunner to Cranberry Point 



145 

18 

53.0 

Ja 






— 


03.1 







180 

00 

00.00 

a' 

Cranberry Point to Gunner 



325 

18 

49.9 





First angle of triangle 

75 

45 

12.8 


O 

44 

t 

50 

n 

44.113 

Gunner 

X 

66 

58 

13. 087 

J<f> 

+ 


04. 529 


JX 

+ 


04. 405 

<t>' 

44 

50 

48. 642 

Cranberry Point 

X' 

66 

58 

17.492 




o r n 

s 

2. 230473 

s 2 

4. 4609 


2 

44 50 46 

COS a 

9. 915025 

sin 2 a 

9. 5103 


n 

B 

8.510480 

C 

1. 4016 


1st term 

-4. 5288 

h 

0. 655978 


5. 3728 

. 

2d term 

0. 0000 






— J(j) 

-4. 5288 



• 



s 

sin a 

A' 

sec p 

2.230473 
9. 755164 
8. 508994 
0.149357 

JX 

sin 

0. 643988 
9. 848315 


0. 643988 


0. 492303 


ft 


tt 

JX 

+4. 4054 

—Ja 

+3.11 

































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. Ill 


secondary triangulation 


STATION GUNNER 








O 

t 

tt 

ct 

Lubec Channel Lighthouse to Duck 


266 

26 

06.5 

Third angle 

Gunner and Duck 



- 31 

13 

30.0 

a 

Lubec Channel Lighthouso to Gimner 


235 

12 

36.5 

Ja 






+ 


17.8 







180 

00 

00.00 

a' 

Gunner to Lubec Channel Lighthouse 


55 

12 

54.3 

<t> 

O 

44 

t 

50 

tt 

31.652 

Lubec Channel 

X 

66 

58 

38.299 





Lighthouse 





d<f> 

+ 


12. 461 


JX 

— 


25.212 

<t>’ 

44 

50 

44.113 

Gimner 

X' 

66 

58 

13.087 




1st term 
2d term 

-J<f> 


44 50 36 


-12.4618 
+ 0.0008 


-12. 4610 


s 

cos a 
B 


2.828793 
9. 756308 
8.510480 


1.095581 


$2 

sin 2 a 
C 


5.6576 
9.8289 
1. 4016 


6.8881 



s 

2.828793 




sin a 

9.914474 




A' 

8.508994 

JX 

1. 401609 


sec <t> 

0.149348 

sin 

9.848294 



1. 401609 


1.249903 



// 


It 


JX 

-25.2121 

—Ja 

-17.78 


STATION CRANBERRY POINT 








o 

9 

ft 

a 

Lubec Channel Lighthouse to Gunner 


235 

12 

36.5 

Third angle 

Cranberry Point and Gunner 


- 14 

08 

48.5 

a 

Lubec Channel Lighthouse to Cranberry Point 

221 

03 

48.0 

Ja 






+ 


14.7 







180 

00 

00.00 

a' 

Cranberry Point to Lubec Channel Lighthouse 

41 

04 

02.7 

<f> 

O 

44 

t 

50 

// 

31. 652 

Lubec Channel 

X 

66 

58 

38.299 




Lighthouse 





J<j> 

+ 


16.990 


JX 

— 


20. 807 

V 

44 

50 

48.642 

Cranberry Point 

X' 

66 

58 

17. 492 




1st term 
2d term 

— J<f> 


O fit 

s 

2.842358 

S“ 

5.6847 

44 50 40 

COS a 

9.877362 

sin 2 a 

9.6350 

It 

B 

8.510480 

C 

1. 4016 

-16.9903 
+ 0.0005 

h 

1.230200 


6.7213 

-16.9898 







s 

sin a 

A' 

sec <t >' 

2.842358 
9.817494 
8. 508994 
0.149357 

JX 

sin 

1.318203 
9.848303 



1.318203 


1.166506 



ft 


It 


JX 

-20.8067 

—Ja 

-14. 67 


91865°—15-8 





























































112 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Preliminary position computation , 

STATION TELEGRAPH 








O 

r 

rr 

a 

Cranberry Point to Lubec Channel Lighthouse 

41 

04 

02.7 

Second angle 

Lubec Channel Lighthouse and Telegraph 


+ 77 

54 

04.2 

a 

Cranberry Point to Telegraph 


118 

58 

06.9 

Jot 






— 


37.4 







180 

00 

00.00 

cc' • 

Telegraph to Cranberry Point 


298 

57 

29.5 





First angle of triangle 

29 

52 

34.9 

<t> 

O 

44 

t 

50 

rr 

48.642 

Cranberry Point 

X 

66 

58 

17.492 

d(f> 

+ 


20.860 


JX 

+ 


52.980 

4>' 

44 

51 

09. 502 

Telegraph 

X' 

66 

59 

10.472 


i( 4 >+ 4 >') 


1st term 
2d term 

— A<f> 


o r rr 

s 

3.123766 

«2 

6.2475 

44 50 59 

COS a 

9.685141 

sin 2 a. 

9. 8839 

rr 

B 

8.510480 

C 

1. 4016 

-20. 8635 
+ 0.0034 

h 

1.319387 


7.5330 

-20.8601 






s 

sin a 
A' 

sec <}>’ 


JX 


3.123766 
9.941951 
8.508994 
0.149401 


1.724112 

rr 

+52.9800 


JX 

sin 


—Joe 


1. 724112 
9. 848343 


1. 572455 


+37. 36 




































APPLICATION OP LEAST SQUARES TO TRIANGULATION. 113 


secondary triangulation —Continued. 

STATION TELEGRAPH 








o 

t 

tt 

a 

Lubec Channel Lighthouse to Cranberry Point 

221 

03 

48.0 

Third angle 

Telegraph and Cranberry Point 

' 

- 72 

13 

20.9 

a 

Lubec Channel Lighthouse to Telegraph 


148 

50 

27.1 

Ja 






— 


22.7 







180 

00 

00.00 

a' 

Telegraph to Lubec Channel Lighthouse 


328 

50 

04.4 


O 

44 

t 

50 

tt 

31.652 

Lubec Channel 

X 

66 

58 

38.299 

j<i> 

+ 


37.850 

Lighthouse 

JX 

+ 


32.173 

V 

44 

51 

09.502 

Telegraph 

X' 

66 

59 

10. 472 


K<£+<£') 


1 st term 
2 d term 

— J4> 


O t ft 

s 

3.135260 

s2 

6.2705 

44 50 50 

COS a 

9.932338 

sin 2 a 

9.4277 

tt 

13 

8.510480 

C 

1. 4016 

-37.8511 
+ 0.0012 

h 

1.578078 


7.0998 

-37.8499 






sm a 
A' 

sec <£' 


JX 


3.135260 
9. 713841 
8.508994 
0.149401 


1. 507496 

“ n 

+32.1733 


JX 

sill £(') 


—Ja 


1. 507496 
9.848324 


1.355820 


+22. 69 




























114 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

FORMATION OF OBSERVATION EQUATIONS 

In the first column of the following table is given the assumed 
azimuths of the various fines. Under station Gunner the azimuth to 
Duck, 299° 38' 04".3, comes from the position computation on 
page 110, and the other azimuths are obtained by adding to this the 
corrected angles from the preliminary computation of triangles. 
Thus in the first triangle in that fist, Gunner-Duck-Larrabee, the 
angle at Gunner is 45° 43' 34."2, which, added to the azimuth of 
Duck, gives 345° 21' 38".5, as shown in the table. For any one 
station the assumed and observed azimuths of some one station 
may be taken as identical. At Gunner they are identical on station 
Duck. The observed azimuths in the second column of the table 
have their symbolic corrections affixed. These azimuths are ob¬ 
tained by adding the observed angles as derived from the fist of 
observed directions on page 106 to the azimuth of the fine Gunner- 
Duck. 

At the fixed stations given in the lower part of the table the method 
of computing the assumed and observed azimuths is somewhat 
different. The assumed azimuths of the fixed fines come from the 
table of fixed positions on page 106. The assumed azimuths of the 
new fines are found by adding to one of these fixed azimuths the 
appropriate corrected angle from the computation of triangles on pages 
107-109. In the second column of the table the observed azimuth 
of one fixed fine used as an initial fine is taken identical with its 
assumed azimuth, and the other observed azimuths, whether of 
fixed fines or of new ones, are found by adding to this azimuth the 
observed angles between the initial fine and each of the others as 
derived from the fist of directions. 

The coefficients of the d<j)’s and &i 7 s are found from the formulas 
on page 93. 


GUNNER 


APPLICATION OF LEAST SQUARES TO TRIANGULATION 


115 


& 

O 

d 

o 


co 


o' 

W 


O 


tid 

S g 

3 a 

W Nl 

< CS 





CT> 

+ 

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C< 

£ 

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CO |H »H 
O IN co —I ^ .X <N 

d<N <x Jcor^co(N + 

t L + 

<< — /</<. << /<C^ 

<"o ‘’o ^ 1 

SS5%§8??^S 

»or^-i^-eocccoioCM 

CO H TP H CO iH rH I 

L L 7 L L+ti 

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<$> 'o «o 'o co 

03 00 OMOiH 
OMHOJlO'rNM 
tOiOf hNOOOh 
X N lO IM N M rt IM 

ttt LULL 

NNNNNNNN 

II II II II II II II II 

ssssssss 


M N N H M N H M 

I I I I I I I I 

cocoi'-cmcmco^os 


CO CM CM rH i—l ICO WO H 

03005000*0r-<<35l0 
COrH^r-criOieMcC'cr' 
OM CO CO • HHH 


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■'I'dooi-H'-j'.-Jr^co 

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05 CO 1-0 CC lO f-1 03 1.0 
05i-i-r>'cfi0CMC0'cr' 
CM CO CO •—I *—< 


H 

g 

M 

o 

Ph 

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pq 

w 

pq 

<< 

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+ 

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s> 


CM 

I 

Cl 

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I 

eo 


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CS o • 00 . 

+d23£«> 

f< + I + + 2, 

fo N N N NT 
Qi /< ✓< e» 

OO'QOh^ 
rH »-H oc »o O 00 
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l^rHrHfHO 

CO<'O< r O<'O«O'0- 
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CO 1-i H 00 -v t'- 

CM S ^ ^ S S3 

I I I I I I 


O ^h Cl CO "f* 

§ S S S S S 


O h N M -»• 

S ^ » S> £> » 

++++++ 

Cl Cl Cl d Cl d 

N N N N N N 

I I I I I I 

ONOOCO^H 


lO iH CO 00 O) 

CS **r i-H co »0 

CO rH tH rH 


Oi t'** o> 




CO rH 1 


B 

pq 

< 

pq 

o 

w 

h-5 

W 

Eh 



H S_4 rJ r-> 


o°° 

vi H' 


82 


CO CO 

05 co cm 
o 00 H' . 

• *"H CO CM 
OHCO® 

+ + + I 

M N r. M 

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cs o> 


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t» O N OO 

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+ + + + 
co eo co eo 
»S t\i fs» bj 

I I I I 

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cs cs’ »o CO 

CS CS rH o 


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TH CS co CO 


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CO 


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CO kO 
CO CO 


337 10 39.4— Zi+vW t?i9=Z4—637<?<£2+1078<5X2+4.3 Cranberry Point 

354 45 17.5— Zi+vW ti 9 '= 24 + 0.0 Lubec church spire 





























MAM 


116 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 




I 

■§ 

g 


M 

.9 

bi 

•S © Opis) S3 g 
M,Q p H 
^333^03 
O^OfitShl 


-tJ 

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o 

Q, © 

fl-2 


p 

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s3 

P 

o 1 

W 



2 t'; 03 ^ O V 

8**£ ++t 

1 $ Ltf ? ii 






II J 1 II i ; 3 

o © p-J © © © 


T3 

© 

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£! 

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Ss & £> & 

++++++ 

<1? <!?<!?<•? 1$ <1? 

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t-~COOCO<N 00 

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£ 

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& 

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fc 




IN 

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+ 

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o 

+ 


t L 

sL~o- 

o§ 

o| 

4-00 
oo 4- 
M L 


S3 

55 


3 3 cl 

55 55 55 

+ + + 

00 cc oo 

M (Vi ^ 

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<n 

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co © 01 co 
+a 

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APPLICATION OF LEAST SQUARES TO TRIANGULATION. 117 


Figure 6 taken with the following two tables shows that z t is for 
directions taken at Gunner, z 2 for directions at Cranberry Point, 
z 3 for directions at Telegraph, and z 5 for directions at Mam. The 
scheme for eliminating these z’s by the use of the sum equations, as 
fictitious additional observation equations with negative weights, 
is used here in the same manner as in the previous example. Each 
weight is the negative reciprocal of the total number of observed 
lines in the adjustment that radiates from the point in question. 
The weights are, respectively, —1/8, —1/6, —1/4, and —1/6, as 
shown in Table 1. 

At Treat,, Larrabee, Indian, and Duck where only one new line 
is to be determined the same process might be used, but the following 
method is identical in results and slightly shorter. Use is made of 
the fact that the directions taken at a point may each be changed 
by the same amount, a change equivalent to using merely a different 
zero point. Correct each of the directions by the averages of all the 
corrections necessary to reduce the observed results to the accepted 
results on the lines that have already been fixed. Then drop the z 
from the observation equation of the new line and assign the equation 


is the number of lines 


a positive weight equal to --, where 

s +1 

already fixed and therefore s +1 is the total number of lines. Thus 
at Larrabee the constant terms of the observation equations repre¬ 


senting pointings on lines already fixed are +2.6, —0.4, +9.0, —0.1, 
and 0.0, the mean of which is +2.8. Subtracting this from each 
of the preceding numbers we have —0.2, —3.2, +6.2, —0.1, and —2.8 
as the new constant terms, also —0.8 instead of +2.0 on Gunner, 
the new point. These new values are inclosed in parentheses and 
are used in forming the normal equations. There are five fixed 
lines, so the weight of the new equation without z that is used to 
replace the six equations containing z is 5/6 and the equation itself is 


v 22 = + 541^-1473^-0.8 


which in Table 2 corresponds to the line No. 22, 


+ 0.49^-1.34^-0.07. 

The z ’s are computed from the sum equations as in the previous 
example, the result of substitution in the right-hand side being di¬ 
vided by —1 /r, r being the total number of lines through the point 
to which z applies. For fixed points where only one new line occurs, 
substitute dc/> and dX in the right-hand side of the observation equation 
on the new line omitting the z, and divide the result by — 1/r. Thus 
at Duck (see p. 116), 

s 8 = (8669^-3509^-1.9) *(- 3) 


as shown in the computation below. 


118 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 

When the z’s are known the v’s or corrections are computed from 
the equations on pages 115 and 116. The details are shown in the 
table on page 121. 

For convenience of solution in the normals it is best to divide the 
constant terms by 10 and the coefficients by 1000. The solution 
will then give 100<^> n and 100^ n . 

Table for formation of normals No. 1 




1 

2 

3 

4 

5 

6 






8<f>i 


8<j> 2 

(JX2 

0<f>3 

5X3 

5? 

V 



z\ 











1 

1 

+ 8669 

- 3509 





+ 

0.0 

1 

1 

2 

1 

+ 2338 

- 1709 





— 

2.2 

1 

1 

3 

1 

+ 541 

- 1473 





+ 

4.8 

1 

1 

4 

1 

- 2191 

- 1388 





+ 

3.6 

1 

1 

5 

1 

- 7756 

- 3833 





+ 

7.1 

1 

1 

6 

1 

- 3643 

+ 1612 



+3643 

-1612 

+ 

9.7 

1 

1 

7 

1 

- 1870 

+ 1580 





+ 

25.5 

1 

1 

8 

1 

-21313 

+21909 

+21313 

-21909 



+ 

7.1 

1 

1 

Sum 

8 

-25025 

+13189 

+21313 

-21909 

+3643 

-1612 

+ 

55.6 

— 8 

0.35355* 


22 






• 





9 

1 

-21313 

+21909 

+21313 

-21909 



+ 

0.0 

1 

1 

10 

1 



- 6013 

- 4910 



+ 

0.0 

1 

1 

11 

1 



- 2010 

- 1485 




13.1 

1 

1 

12 

1 



- 4189 

+ 1650 

+4189 

—1650 

_ 

2.7 

1 

1 

13 

1 



- 2045 

+ 1701 



+ 

11.5 

1 

1 

14 

1 



- 637 

+ 1078 



+ 

6.8 

1 

1 

Sum 

6 

-21313 

+21909 

+ 6419 

-23875 

+4189 

-1650 

+ 

2.5 

i 

0.4083* 

15 

23 











1 





-1181 

+4922 

+ 

0.0 

1 

1 

16 

1 



- 4189 

+ 1650 

+4189 

-1650 

— 

52.6 

1 

1 

17 

1 

- 3643 

+ 1612 



+3643 

-1612 

— 

54.8 

1 

1 

18 

1 





+2413 

-2S39 

— 

62.2 

1 

1 

Sum 

4 

- 3643 

+ 1612 

- 4189 

+ 1650 

+9064 

-1179 

-169.6 

-i 

0.5* 

19 

25 



- 637 

+ 1078 



+ 

4.3 

i 

0.707 

20 

1 



- 2010 

- 1485 



+ 

6.4 

1 

1 

21 

1 

- 2191 

- 1388 





+ 

1.1 

1 

1 

Sum 

6 

— 2191 

- 1388 

- 2010 

- 1485 



+ 

7.5 

-i 

0.4083* 

22 


+ 541 

- 1473 






0.8 

5 

0.9129 

23 


+ 2538 

- 1709 





_ 

2.8 

4 

0. 8944 

24 


+ 8669 

- 3509 





— 

1.9 

2 

¥ 

0. 8165 

























APPLICATION OF LEAST SQUARES TO TRIANGULATION. 119 


Table for formation of normals No. 2 



Hi 

<JXi 


5X a 

Hi 

ax 3 


I 

1 

+ 8.67 

- 3.51 





+0.00 

+ 5.16 

2 

+ 2.54 

- 1.71 





-0.22 

+ 0.61 

3 

+ 0.54 

- 1.47 





+0.48 

- 0.45 

4 

- 2.19 

- 1.39 





+0.36 

- 3.22 

5 

- 7.76 

- 3.83 





+0.71 

-10.88 

6 

- 3.64 

+ 1.61 



+3.64 

-1.61 

+0.97 

+ 0.97 

7 

- 1.87 

+ 1.58 





+2.55 

+ 2.26 

8 

-21.31 

+21.91 

+21.31 

-21.91 



+0.71 

+ 0.71 

Zl 

- 8.85i 

+ 4.66i 

+ 7.541 

- 7.751 

+1.291 

-0.571 

+1.971 

- 1.711 

9 

—21.31 

+21.91 

+21.31 

-21.91 



+0.00 

+ 0.00 

10 



- 6.01 

- 4.91 



+0.00 

-10.92 

11 



- 2.01 

- 1.48 



-1.31 

- 4.80 

12 



- 4.19 

+ 1.65 

+4.19 

-1.65 

-0.27 

- 0.27 

13 



- 2.04 

+ 1.70 



+ 1.15 

+ 0.81 

14 



- 0.64 

+ 1.08 



+0.68 

+ 1.12 

za 

— 8.70i 

+ 8.95i 

+ 2.621 

- 9.751 

+1.711 

-0.671 

+0.101 

- 5.741 

15 





-1.18 

+4.92 

+0.00 

+ 3.74 

16 



- 4.19 

+ 1.65 

+4.19 

-1.65 

-5.26 

- 5.26 

17 

- 3.64 

+ 1.61 



+3.64 

-1.61 

-5.48 

- 5.48 

18 





+2.41 

-2.84 

-6.22 

- 6.65 

z 3 

- 1.82i 

+ 0.81i 

- 2.091 

+ 0.821 

+4.531 

-0.591 

-8.481 

- 6.821 

19 



- 0.45 

+ 0.76 



+0.30 

+ 0.61 

20 



- 2.01 

- 1.48 



+0.64 

- 2.85 

21 

- 2.19 

— 1.39 





+0.11 

- 3.47 

Z5 

- 0.89i 

- 0.57 i 

- 0.821 

- 0.611 



+0. 311 

- 2.581 

22 

+ 0.49 

- 1.34 





-0.07 

- 0.92 

23 

+ 2.27 

- 1.53 





-0.25 

+ 0.49 

24 

+ 7.08 

- 2.86 





-0.16 

+ 4.06 


Normal equations 


1 

2 

3 

4 

5 

6 

i} 

1 

+987.3526 

-852.5451 
+913.2320 

-823.2428 
+876.4443 
+923.5619 

+781.3412 
-837.7306 
-831.4801 
+842.4946 

+ 8.0389 
-13.2644 
-39.8513 
+36.7824 
+43.7028 

- 0.2265 
+ 3.9464 
+18.6471 
-15.9112 
-33.6441 
+41.7793 

- 8.9085 
+ 6.5262 
+ 4.1465 
+ 2.6136 
-18. 8752 
+30.2371 

+ 91.8098 
+ 96.6088 
+ 128.2256 

- 21.8901 

- 17.1109 
+ 44.8281 























Solution of normals 


120 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 



+ 91. 8098 
- 0.0929858 

+ 96.6088 
+ 79.2746 

+ 175. 8834 
- 0.9931944 

+128.2256 
+ 76.5499 
-164.4754 

+ 40.3001 
- 0.489729 

- 21.8901 

- 72.6536 

+ 161.9595 

+ 13.4747 

+ 80.8905 

- 1.247908 

- 17.1109 

- 0.7475 

+ 6.2801 

+ 13.3381 

- 19.3324 

- 17.5726 

+ 0.572491 

+ 44.8281 

+ 0.0211 

- 3.7253 

- 7.3218 

+ 9.0837 

- 15.3544 

+ 27.5314 

- 1.86892 

s* 

- 8.9085 
+ 0.0090226 

+ 6.5262 

- 7.6922 

- 1.1660 

+ 0.0065843 

+ 4.1465 

- 7.4278 
+ 1.0904 

- 2.1909 
+ 0.026624 

+ 2.6136 
+ 7.0497 

- 1.0737 

- 0. 7326 

+ 7.8570 

- 0.121211 

-18.8752 

+ 0.0725 

- 0.0416 

- 0. 7251 

- 1.8778 

—21. 4472 

+ 0.698720 

+30. 2371 

- 0.0020 

+ 0.0247 

+ 0.3980 

+ 0.8823 

-18.7399 

+ 12.8002 

- 0.86892 


- 0.2265 
+ 0.0002294 

+ 3.9464 

- 0.1956 

+ 3.7508 

- 0.0211804 

+ 18.6471 

- 0.1889 

- 3.5075 

+ 14.9507 

- 0.181682 

-15.9112 
+ 0.1792 
+ 3.4539 
+ 4.9989 

- 7.2792 
+ 0.112297 

-33.6441 
+ 0.0019 
+ 0.1339 
+ 4.9482 
+ 1.7397 

-26.8204 
+ 0.873771 

+41. 7793 

- 0.0001 

- 0.0794 

- 2.7163 

- 0.8174 

-23. 4349 

+ 14.7312 

3k 


+ 8.0389 
- 0.0081419 

-13.2644 
+ 6.9413 

- 6.3231 
+ 0.0357059 

-39. 8513 
+ 6.7027 
+ 5.9130 

-27. 2356 
+ 0.330969 

+36. 7824 

- 6.3616 

- 5.8225 

- 9.1065 

+ 15. 4918 

- 0.238994 

+43. 7028 

- 0.0655 

- 0.2258 

- 9.0141 

- 3. 7024 

+30.6950 

1 

2 

3 

4 

5 

■*« 

+ 781.3412 
- 0.7913497 

-837. 7306 
+674.6613 

-163.0693 
+ 0.9208345 

-831. 4801 
+651. 4730 
+ 152.4924 

- 27.5147 
+ 0.334360 

+842. 4946 
-618.3141 
-1.50.1598 
- 9.1998 

+ 64.8209 

1 

2 

3 

4 

CO 

-823. 2428 
+ 0.8337881 

+876.4443 
-710.8419 

+ 165.6024 
- 0.9351387 

+923. 5619 
-686. 4101 
-154.8612 

+ 82.2906 
$<t>2 

1 

2 

3 

\ * • 


-852. 5451 
+ 0.8634657 

+913. 2320 
-736.1434 

+ 177.0886 

dAi 

1 

2 


CO T—1 

<N 

CO 

+ 























APPLICATION OF LEAST SQUARES TO TRIANGULATION 


121 


i Bach solution 




d<f>3 

5A 2 

8<t> 2 

dki 

Hi 


-0.86892 

+0.69872 
-0.75924 

-0.12121 
-0.09758 
+0.01446 

+0.02662 
+0.15787 
-0.02003 
-0.06832 

+0.00658 
+0.01840 
-0.00216 
-0.18815 
-0.08990 

+0.00902 
-0.00020 
+0.00049 
+0.16170 
+0.08016 
-0.22038 

-0.86892 

-0.06052 

-0.20433 

+0.09614 

-0.25523 

+0.03079 

♦ 


Computation of corrections 

1 

2 

3 

4 

5 

6 

7 

8 

+ 2.669 
+ 8.956 
-11.412 

+ 0.781 
+ 4.362 
-11.412 
- 2.2 

+ 0.167 
+ 3.760 
-11.412 
+ 4.8 

- 0.675 
+ 3.543 
-11.412 
+ 3.6 

- 2.388 
+ 9.783 

- 11.412 
+ 7.1 

- 1.122 

- 4.114 

- 2.205 
+ 14.007 
-11.412 
+ 9.7 

- 0.576 

- 4.033 
-11.412 
+25.5 

- 6.562 
-55.918 
+20.490 
+44. 767 
-11.412 
+ 7.1 

+ 0.213 
+ 0.2 

- 8.469 

- 8.5 

- 2.685 

- 2.7 

- 4.944 

- 4.9 

+ 3.083 
+ 3.1 

+ 9.479 
+ 9.5 

+ 4.854 
+ 4.9 

- 1.535 

- 1.5 

Zi 

9 

10 

11 

12 

13 

14 

Z2 

- 7.705 
-33.662 
+20.490 
+44.767 

- 2.205 
+ 14.007 
+55.6 

- 6.562 
-55.918 
+20.490 
+44.767 

- 1.129 

- 5.781 
+ 10.033 

- 1.129 

- 1.932 
+ 3.035 

- 1.129 
-13.1 

- 4.027 

- 3.371 

- 2.535 
+ 14.337 

- 1.129 

- 2.7 

- 1.966 

- 3.476 

- 1.129 
+ 11.5 

- 0.612 

- 2.203 

- 1.129 
+ 6.8 

- 6.562 
-55.918 
+ 6.171 
+48.783 

- 2.535 
+ 14.337 
+ 2.5 

+ 3.123 
+ 3.1 

-13.126 

-13.1 

+ 4.929 
+ 5.0 

+ 2.856 
+ 2.9 

+ 1.648 
+ 1.7 

+ 0.575 
+ 0.6 

+91.292 

-11.412 

+ 6.776 
- 1.129 

15 

16 

17 

18 

23 

19 

24 

19' 

+ 0.715 
-42. 768 
+44.369 

- 4.027 

- 3.371 

- 2.535 
+ 14.337 
+44.369 
-52.6 

- 1.122 

- 4.114 

- 2.205 
+ 14.007 
+44.369 
-54.8 

- 1.460 
+24.669 
+44.369 
-62.2 

- 1.122 

- 4.114 

- 4.027 

- 3.371 

- 5.486 
+ 10.245 
-169.6 

- 0.612 

- 2.203 

- 0.742 
+ 4.3 

- 0.612 
- 2.203 
+ 4.3 

- 0.742 

- 0.742 

- 0.7 

+ 2.316 
+ 2.3 

+ 1.485 
- 0.742 

+ 5.378 
+ 5.4 

+ 0.743 
+ 0.8 

- 3.827 

- 3.8 

- 3.865 

- 3.8 

-177.475 
+ 44.369 

20 

21 

25 

20' 

20" 

20'" 

20"" 

22 

- 1.932 
+ 3.035 

- 1.912 
+ 6.4 

- 0.675 
+ 3.543 

- 1.912 
+ 1.1 

- 0.675 
+ 3.543 

- 1.932 
+ 3.035 
+ 7.5 

- 1.912 
+ 1.1 

- 1.912 
+ 0.4 

- 1.912 

- 4.6 

- 1.912 
+ 3.1 

+ 0.167 
+ 3.760 

- 0.521 

- 0.8 

- 0.812 
- 0.8 

- 1.512 

- 1.5 

- 6.512 

- 6.5 

+ 1.188 
+ 1.2 

+ 5.591 
+ 5.6 

+ 2.056 
+ 2.1 

+ 2.606 
+ 2.6 

+11.471 
- 1.912 

26 

22' 

22" 

22'" 

22"" 

22 v 

23 

27 

+ 0.167 
+ 3.760 
- 0.8 

- 0.521 

- 0.2 

- 0.521 

- 3.2 

- 0.521 
+ 6.2 

- 0.521 

- 0.1 

- 0.521 

- 2.8 

+ 0.781 
+ 4.362 

- 0.469 

- 2.8 

+ 0.781 
+ 4.362 
- 2.8 

- 0.721 

- 0.7 

- 3.721 

- 3.7 

+ 5.679 
+ 5.7 

- 0.621 

- 0.6 

- 3.321 

- 3.3 

+ 3.127 
- 0.521 

+ 2.343 
- 0.469 

+ 1.874 
+ 1.9 

23' 

23" 

23'" 

23"" 

24 

28 

24' 

24" 

- 0.469 

- 3.8 

- 0.469 

- 5.0 

- 0.469 

- 2.4 

- 0.469 
+ 11.2 

+ 2.669 
+ 8.956 

- 3.242 

- 1.9 

+ 2.669 
+ 8.956 
- 1.9 

- 3.242 
+ 0.2 

- 3.242 

- 0.1 

- 4.269 

- 4.2 

- 5.469 

- 5.5 

- 2.869 

- 2.8 

+ 10.731 
+ 10.7 

- 3.042 

- 3.0 

- 3.342 

- 3.3 

+ 9.725 
- 3.242 

+ 6.483 
+ 6.5 























































































































122 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final computation of triangles 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

Spher 

ical 

excess 

Plane angle 

Loga¬ 

rithm 



O 

f 

// 

// 

n 


O 

/ 

// 



Duck-Larrabee 










3.410111 

-1+ 3 

Gunner 

45 

43 

29.4 

- 2.9 

26.5 





0.145096 

-24'+24 

Duck 

124 

02 

01.1 

+ 9.5 

10.6 





9.918389 

-22+22"" 

Larrabee . 

10 

14 

26.1 

- 3.2 

22.9 





9.249850 






+ 3.4 






/ 


Gunner-Larrabee 










3.473596 


Gunner-Duck 










2.805057 


Duck-Mam 










3.389282 

-1+ 4 

Gunner 

108 

41 

53.9 

- 5.1 

48.8 





0.023546 


Duck 



58.8 


67.5 


57 

00 

67.5 

9.923684 

-21+20" 

Mam 

14 

17 

07.3 

- 3.6 

03.7 





9.392230 


Gunner-Mam 










3.336512 


Gunner-Duck 










2.805058-1 


Duck-Lubec Chan- 










3.045619 


nel Lighthouse 











— 1+ 5 

Gunner 

115 

34 

42.9 

+ 2.9 

45.8 





0.044799 

—24"+24 

Duck 

33 

11 

41. S 

+ 9.8 

51.6 





9.738407 


Lubec Channel 



35.3 


22.6 


31 

13 

22.6 

9.714639 


Lighthouse 
Gunne r-L u b e c 










2.828825+2 


Channel Light¬ 
house 












Gunner-Duck 










2.805057 


Indian Point-Lar- 










3.236668 


rabee 











-2+ 3 

Gunner 

31 

57 

41.4 

+ 5.8 

47.2 





0.276238 

—23'+23 

Indian Point 

114 

00 

37.5 

+ 6.1 

43.6 





9.960689 

—22+22 v 

Larrabee 

34 

01 

35.1 

- 5.9 

29.2 





9.747840 






+ 6.0 








Gunner-Larrabee 










3.473595+1 


Gunner- Indian 










3.260746 


Point 












Indian Point-Mam 










3.470097 

—2+ 4 

Gunner 

94 

56 

05.9 

+ 3.6 

03.5 





0.001614 

—23"+23 

Indian Point 

47 

05 

36.5 

+ 7.4 

43.9 





9.864802 

-21+20'" 

Mam 

37 

58 

15.2 

- 8.6 

06.6 





9.789036 






+ 2.4 








Gunner-Mam 










3.336513-1 


Gunner- Indian 










3.260747-1 


Point 












Indian Point-Lu- 










3.315762 


bee Channel 
Lighthouse 











-2+ 5 

Gunner 

1C1 

48 

54.9 

+ 11.6 

66. 5 





0.009305 

—23"'+23 

Indian Point 

18 

35 

59.1 

+ 4.7 

63.8 





9.503759 

Lubec Channel 



66.0 


49.7 


59 

34 

49.7 

9.935679 


Lighthouse 
Gunne r-L u b e c 










2.828826+1 


Channel Light¬ 
house 












Gunner- Indian 










3.260746 


Point 












Larrabee-Mam 










3.443126 

-3+ 4 

Gunner 

62 

58 

24.5 

- 2.2 

22.3 





0.050224 

-22 '+22 

Larrabee 

44 

10 

36.5 

+ 3.3 

39.8 





9.843162 

-21+20"" 

Mam 

72 

50 

58.8 

- 0.9 

57.9 





9.980246 






+ 0.2 








Gunner-Mam 










3.336512 


Gunner-Larrabee 










3.473596 


Larr abee-Lubec 










3.449573 


Channel Light¬ 
house 











-3+ 5 
—22"+22 

Gunner 

G9 

51 

13.5 

+ 5.8 

19.3 





0.027415 

Larrabee 

12 

59 

26.4 

+ 0.3 

32.7 





9.351839 


Lubec Channel 



20.1 


08.0 


97 

09 

08.0 

9.996608 


Lighthouse 

G unner-Lubec 










2.828827 


Channel Light¬ 
house 












Gunner-Larrabee 










3.473596 


































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 123 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

Spher¬ 

ical 

excess 

Plane angle 

Loga¬ 

rithm 



O 

/ 

n 

ft 

// 


O 

/ 

II 



Larra b e e-L ubec 










3.702872 


church spire 











-3+ 7 

Gunner 

154 

32 

48. 4 

+ 12.2 

60.6 





0.366814 

—22"'+22 

Larrabee 

10 

46 

05.9 

- 3.1 

02.8 





9.271431 

Lubec church spire 



65.7 


56.6 


14 

40 

56.6 

9.403910 


G u n n e r-L ubec 










3.341117 


church spire 
Gunner-Larrabeo 










3.473596 


Mam-Lubec Chan- 










3.176968 


nel Lighthouse 











-4+ 5 

Gunner 

6 

52 

49.0 

+ 8.0 

57.0 





0.921421 

—20'+21 

Mam 

3 

04 

50.7 

+ 2.9 

53.6 





8.730438 

Lubec Channel 



20.3 


09.4 


170 

02 

09.4 

9.238122 


Lighthouse 
Gunne r-L ubec 










2.828827 


Channel Light¬ 
house 












Gunner-Mam 










3.336511+ 1 


Lubec church spire- 










3.603122 

1 

<N 

+ 

Indian Point 
Gunner 

173 

29 

30.2 

-18.0 

12.2 



57 


0.945259 

Lubec church spire 



11.8 


21.0 


2 

21.0 

8. 712365 

-23+ 23"" 

Indian Point 

3 

33 

18.0 

+ 8.8 

26.8 





8.792736 

Gunner-Ind ian 










3.260746 


Point 

Gunne r-L ubec 










3.341117 


church spiro 












Gunne r-L ubec 










2.828827 


Channel Light¬ 
house 











—9+10 

Cranberry Point 

75 

45 

12.8 

+ 1.4 

14.2 





0.013565 

-5+ 8 

Gunner 

90 

05 

58.7 

— 4.6 

54.1 


14 

08 

51.7 

9.999999 

Lubec Channel 



48.5 


51.7 


9.388141 


Lighthouse 
Cranberry Point- 










2.842391 


Lubec Channel 












Lighthouse 
Cranberry Point- 










2.230533+ 1 


Gunner 












Gunner-Mam 




-14.8 

51.1 





3.336512 

—9+11 

Cranberry Point 

78 

36 

65.9 





0.008632 

—4+ 8 

Gunner 

96 

58 

47.7 

+ 3.4 

51.1 





9.996769 

-20+21 

Mam 

4 

24 

21.3 

- 3.5 

17.8 





8.885390 






-14.9 








Cranberry Point- 










3.341913 


Mam 

Cranberry Point- 










2.230534 


Gunner 












Gunne r-L ubec 










3.341117 

-9+13 
-7+ 8 

church spire 
Cranberry Point 
Gunner 

174 

5 

08 

24 

32.3 

23.8 

+ 3.28 
-11.02 

35.58 
12.78 


0 

27 

11.64 

0.991220 

8.973913 

Lubec church soire 



03.9 


11.64 


7.898195 


Cranberry Point- 










3.306250 


Lubec church 












spire 

Cranberry Point- 










2.230532+ 2 


Gunner 












Lubec Channel 










3.176968 


L ighthouse- 











-10+11 

Mam 

Cranberry Point 
Lubec Channel 

2 

51 

53.1 

36.3 

-16.2 

36.9 

58.9 


175 

48 

58.9 

1.301899 

8.863046 

-20+20' 

Lighthouse 

Mam 

Cranberry Point- 

1 

19 

30.6 

- 6.4 

24.2 





8.363526 
3.341913 


Mam 

Cranberry Point- 






• 




2.842339-2 


Lubec Channel 
Lighthouse 






































124 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 . 


Symbol 


-13+14 

-19+19' 


-15+16 

-12+13 


-15+17 

-6+7 


-16+17 
- 9+12 
- 6+8 


-16+18 

- 10+12 


-17+18 

-5+6 


Final computation of triangles —Continued 


Station 


Lubec church spire- 
Treata 

Cranberry Point 
Lubec church spire 
Treats 

Cranberry Point- 
Treats 

Cranberry Point- 
Lubec church 
spire 

Lubec church 
s p i re-Cranberry 
Point 
Telegraph 
Lubec church spire 
Cranberry Point 
T e 1 e g r aph-Cran- 
berry Point 
Telegraph-L u b e c 
church spire 

Lubec church 
spire-Gunner 
Telegraph 
Lubec church spire 
Gunner 

Telegraph-Gunner 
Telegraph-L u b e c 
church spire 

Cranberry Point- 
Gunner 
Telegraph 
Cranberry Point 
Gunner 


Telegraph-Gunner 
T e 1 e g r aph-Cran- 
berry Point 

Cranberry Point- 
Lubec Channel 
Lighthouse 
Telegraph 
Cranberry Point 
Lubec Channel 
Lighthouse 
Telegraph-L u b e c 
Channel Light¬ 
house 

T e 1 e g r aph-Cran- 
berry Point 

Gunner-Lubec 
Channel Light¬ 
house 
Telegraph 
Gunner 

Lubec Channel 
Lighthouse 
Telegraph-L u b e c 
Channel Light¬ 
house 

Telegraph-Gunner 


Observed 

angle 

Correc¬ 

tion 

Spheri¬ 

cal 

angle 

Spher¬ 

ical 

excess 

Plane angle 

Loga¬ 

rithm 

o 

/ 

ft 

// 

tf 


0 

/ 

// 

3.309982 

17 

44 

02.9 

- 2.1 

00.8 





0.516283 

19.0 


22.6 


144 

41 

22.6 

9.761932 

17 

34 

38.1 

- 1.5 

36.6 





9.479985 
3.588197 










3.306250 










3.306250 

128 

38 

59.9 

- 6.1 

53.8 




49.2 

0.107352 

47.5 


49.2 


30 

51 

9.710115 

20 

29 

12.6 

+ 4.4 

17.0 





9.544083 

3.123717 




. 






2.957685-J 










3.341117 

131 

33 

53.1 

- 6.1 

47.0 





0.125967 

36.1 


37.6 


30 

24 

37.6 

9.704315 

18 

01 

30.8 

+ 4.6 

35.4 





9.490600 







3.171399 
2.957684 













2.230534 

2 

54 

53.2 

0.0 

53.2 





1.293705 

153 

39 

19.7 

- 1.1 

18.6 





9.647160 

23 

25 

54.6 

- 6.4 

48.2 





9.599478 




- 7.5 




- 


3.171399 

3.123717 










2.842391 

29 

52 

44.5 

+ 9.3 

53.8 





0.302588 

77 

54 

06.9 

- 2-5 

04.4 





9.990245 



08.6 


01.8 


72 

13 

01.8 

9.978738 










3.135224 










3.123717 










2.828827 

26 

57 

51.3 

+ 9.2 

60.5 





0.343447 

66 

40 

04.1 

+ 1.8 

05.9 





9.962950 



64.6 


53.6 


86 

21 

53.6 

9.999125 





• 





3.135224 










3.171399 

















APPLICATION OF LEAST SQUARES TO TRIANGULATION. 125 
SECOND METHOD. 

The only difference between the second method of adjustment and 
the first is in the treatment of the directions taken at the fixed points. 
At these points observation equations are written for the directions 
of new points only and the z’s are omitted. The observations taken 
over the fixed lines are not used, but the observed directions of the 
new lines are taken in connection with the adjusted direction of a 
fixed fine, all directions being referred to a common initial line. 

The equations with z’s omitted are the same as if the angle method 
of adjustment were used. (See p. 196.) In this treatment these equa¬ 
tions are given unit weight. Jordan (Vermessungskunde, vol. 1, p. 
179, of the third edition) suggests that on some accounts it would be 
better to assign the equations for observations at fixed points only 
half weight. 

The observation equations for directions taken at Gunner, Cran¬ 
berry Point, and Telegraph are the same as for the first method 
given on page 115 and are not repeated here. Below are given the 
observation equations for the remaining points, formed according to 
the second method. The assumed azimuths are identical with those 
used in the first method. As an examplo, to illustrate the computa¬ 
tion of the observed azimuths, take the fine Mam-Gunner. Use the 
observed direction for the new line and the adjusted one for the fixed 
line. 

o / // 

Fixed azimuth Mam to Indian Point, page 106, =266 17 19. 2 

Angle Indian Point to Gunner, page 106 (359° 

59' 55".4 to 322° OF 44 // .8), =322 01 49.4 

Observed azimuth Mam to Gunner, =228 19 08. 6 

or by reckoning from any other fixed line through Mam the same 
result is reached, thus, 

Fixed azimuth Mam to Lubec Channel Light- ° ' n 

house, =225 14 19.0 

Angle Lubec Channel Lighthouse to Gunner, 
page 106 (318° 56' 55"2 to 322° OF 44"8), = 3 04 49.6 

Observed azimuth Mam to Gunner, =228 19 08. 6 

Note that the coefficients of the Jt^s and JA’s are exactly the same 
as for the first method and that the z’s are omitted. 




126 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


/ 


eo 

£ 

O 

HO 

i 

Ch 


si 



EH 

< 

W 

« 

&H 


'O 

> 


.O 

O 


S3 

.2 

+-> 

cS 

+5 

02 


.a 


o 


PH 


S3 

.2 

•p 

oJ 

S3 

CH 

w 


+ 


+ 



t- 



Cranberry Point 

Gunner 

LARRABEE 

Gvuuier 

Tf« t-H 

CD 7—5 

++ 

©4 r-4 

/< ^ 

**© 
io oo 

00 00 
rf CO 

t-H t-H 

1 1 

©1 ^ 
•e--©. 

O rH 

7-H 

O T-H 

CS CS 

1 1 

II II 

©* ©3 

t-H 

t-H 

I 
< 

'O 

g 

Tf< 

rH 

1 

-©• 

rH 

Tji 

lO 

+ 

II 
©» 

©1 

O —t 
©J ©4 

+ + 

CO co 

r-* oo 
^ o 

C5 

Tf t-H 

CO 00 

CS CS 

CS CS 

©4 

M 

+ 

r- 

CO 

o 

CS 

CS 

iO 

CO 

rH 



CO 

s'g 


CS 

o 

^ a> 


CS 

*rp r-H 


CS 

CO oo 


»o 

CS CS 


CO 

<N <M 


t-H 


A 

£ 

i—i 

O 

PH 

fc 

M 

A 

fc 



>« 

<N 

+ 

«o 


I 


i 

<N 

+ 


+ 


S3 

7—4 


9 


9 




W 

o 

A 

A 


119 38 24.0+^24 t’ 24 = +8669^1—35095X1—1.9 Gunner 

























APPLICATION OF LEAST SQUARES TO TRIANGULATION. 127 

The first part of each of tables 1 and 2 for the formation of normals 
according to the first method, pages 118 and 119, down to the line for 
v 1Q will serve for the second method also and is not repeated here. 
The remainder of the tables according to the second method is given 
below. 

In forming the normal equations the four z ’s that occur are elimi¬ 
nated by the device of the sum equation serving as a fictitious obser¬ 
vation equation with negative weight. The other observations that 
do not contain z’s enter into the formation of the normal equations 
in the usual way. After the normal equations have been solved the 
four z’s are found from the sum equations in tho way previously 
explained and enter into tho computation of the v’s, or corrections, 
from 1 to 18 and 20 and 21, hut not into the others. 


'Table for formation of normals No. 1 



1 

2 

3 

4 

5 

6 





. Hi 

8 k 

8 fa 

8 X 2 

8 fa 

8 k 3 

V 

V 

Vp 

19 



- 637 

+ 1078 



+ 

4.3 

1 

1 

20 



- 2010 

- 1485 



+ 

6.4 

1 

1 

21 

- 2191 

- 1388 





+ 

1.1 

1 

1 

22 

+ 541 

- 1473 






1.1 

1 

1 

23 

+ 2538 

- 1709 





+ 

2.5 

1 

1 

24 

+ 8669 

- 3509 






1.9 

1 

1 


Table far formation of normals No. 2 



1 

2 

3 

4 

5 

6 




8 fa 

5Xi 

8 fa 

8 M 

8 fa 

8 \3 

V 

1 

19 



- 0.64>, 

+ 1.08 



+0.43 

+ 0.87 

20 



- 2.01 

- 1.48 



+0.64 

- 2.85 

21 

- 2.19 

- 1.39 





+0.11 

- 3.47 

22 

+ 0.54 

- 1.47 





-0.11 

- 1.04 

23 

+ 2.54 

- 1.71 





+0.25 

+ 1.08 

24 

+ 8.67 

- 3.51 





-0.19 

+ 4.97 


Normal equations 


1 

2 

3 

4 

5 

6 

V 

1 

+1014.5374 

-863.2282 

+918.6459 

-822.5130 
+876.9117 
+924. 4414 

+781.8841 
-837.3829 
-831.3291 
+843.4555 

+ 8.0389 
-13.2644 
-39.8513 
+36.7824 
+43.7028 

- 0.2265 
+ 3.9464 
+18.6471 
-15.9112 
-33.6441 
+41. 7793 

- 8.5215 
+ 5.8167 
+ 3.7521 
+ 2.6609 
-18.8752 
+30.2371 

+109.9712 
+ 91.4452 
+130.0589 

- 19.8403 

- 17.1109 
+ 44.8281 


91865°—15-9 





































Solution of normals 


128 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 



+ 109. 9712 
- 0.1083954 

+ 91.4452 
+ 93.5700 

+185.0152 
- 1.004641 

1 +130.0589 

+ 89.1566 
-177. 8910 

+ 41.3245 
- 0.473054 

- 19.8403 

- 84.7526 

+172.9087 

+ 15.1150 

+ 83.4306 

- 1.220852 

- 17.1109 

- 0.8714 

+ 6.4543 

+ 12.8467 

- 17.8855 

- 16.5668 

+ 0.520443 

+ 44.8281 

+ 0.0246 

- 3.7712 

- 7.0269 

+ 8.2961 

- 14.2794 

+ 28.0712 

- 1.89000 


- 8.5215 
+ 0.0083994 

+ 5.8167 

- 7.2506 

- 1.4339 
+ 0.007786 

+ 3.7521 

- 6.9086 
+ 1.3787 

- 1.7778 
+ 0.020351 

+ 2.6609 
+ 6.5674 

- 1.3401 

- 0.6503 

+ 7.2379 

- 0.105913 

-18. 8752 

+ 0.0675 

- 0.0500 

- 0.5527 

- 1.5516 

-20.9620 

+ 0.658518 

+30. 2371 

- 0.0019 

+ 0. 0292 

+ 0.3023 

+ 0.7197 

-18.0677 

+ 13. 2187 

- 0.89000 

CO 

- 0.2265 
+ 0.0002233 

+ 3.9464 

- 0.1927 

+ 3.7537 

- 0.020383 

+18.6471 

- 0.1836 

- 3.6092 

+14.8543 
( - 0.170042 

-15.9112 
+ 0.1746 
+ 3.5081 
+ 5.4332 

- 6.7953 
+ 0.099437 

-33.6441 
+ 0.0018 
+ 0.1309 
+ 4.6178 
+ 1.4567 

-27. 4369 
+ 0.861926 

+41. 7793 

- 0.0001 

- 0.0765 

- 2.5259 

- 0.6757 

-23.6486 

• +14.8525 

8X3 


+ 8.0389 
- 0.0079237 

-13.2644 
+ 6.8400 

- 6.4244 
+ 0.034885 

-39. 8513 
+ 6.5174 
+ 6.1770 

-27.1569 
+ 0.310873 

+36. 7824 

- 6.1954 

- 6.0040 

- 9.9330 

+14.6500 

- 0.214376 

+43. 7028 

- 0.0637 

- 0. 2241 

- 8.4423 

- 3.1406 

+31. 8321 
d<f >2 

1 

2 

3 

4 

5 


+781. 8841 
- 0.7706804 

-837.3829 
+665.2730 

-172.1099 
+ 0.934565 

-831.3291 
+633.8946 
+165.4826 

- 31.9519 
+ 0.365763 

+843. 4555 
-602. 5828 
-160.8479 
- 11.6868 

+ 68.3380 

8 X 2 

1 

2 

3 

4 

CO 

-822.5130 
+ 0.8107271 

+876.9117 
-699. 8425 

+177.0692 
- 0.961494 

+924. 4414 
-666.8336 
-170. 2510 

+ 87.3568 

8 <f> 2 

1 

2 

3 


-863. 2282 
+ 0.8508589 

+918.6459 
-734. 4854 

+184.1605 
<5Ai 

1 

2 


+1014. 5374 
8<j> i 

l 

























APPLICATION OF LEAST SQUARES TO TRIANGULATION. 129 


Back solution 



3 

<U a 

8<f> 2 

9Xi 

S<f>i 

-0.89000 

+0. 65852 
-0. 76711 

-0.10591 
-0.08850 
+0.02328 

+0.02035 
+0.15131 
-0.03376 
-0.06259 

+0.00779 
+0.01814 
-0.00379 
-0.15993 
-0.07244 

+0.00840 
-0.00020 
+0.00086 
+0.13189 
+0.06108 
-0.17888 

-0.89000 

-0.10859 


-0.17113 



+0.07534 




-0.21023 





+0.02315 


Computation of corrections 


1 

2 

3 

4 

5 

6 

7 

8 

+ 2.007 
+ 7.377 
-10. 753 

- 1.369 

- 1.4 

+ 0.588 
+ 3.593 
-10.753 
- 2.2 

+ 0.125 
+ 3.097 
-10.753 
+ 4.8 

- 0.507 
+ 2.918 
-10.753 
+ 3.6 

- 1.796 
+ 8.058 
-10.753 
+ 7.1 

- 0.843 

- 3.389 

- 3.956 
+14. 347 
-10.753 
+ 9.7 

- 0.433 

- 3.322 
-10.753 
+25.5 

- 4.934 
-46.059 
+ 16.057 
+37. 493 
-10.753 
+ 7.1 

- 8.772 

- 8.8 

- 2.731 

- 2.8 

- 4.742 

- 4.8 

+ 2.609 
+ 2.6 

+ 10.992 
+ 11.0 

+ 5.106 
+ 5.1 

- 1.096 

- 1.1 

Zl 

9 

10 

11 

12 

13 

14 

*2 

- 5.793 
-27. 727 
+ 16.057 
+37. 493 

- 3.956 
+ 14. 347 
+55.6 

- 4.934 
' -46.059 

+ 16.057 
+37. 493 

- 1.223 

+ 1.334 
+ 1.3 

- 4.530 
+ 8.402 

- 1.223 

+ 2.649 
+ 2.7 

- 1.514 
+ 2.541 

- 1.223 
-13.1 

- 3.156 

- 2.824 

- 4.549 
+ 14.685 

- 1.223 
-2.7 

+ 0.233 
+ 0.2 

- 1.541 

- 2.911 

- 1.223 
+ 11.5 

- 0.480 

- 1.845 

- 1.223 
+ 6.8 

- 4.934 
-46.059 
+ 4.836 
+40. 857 

- 4.549 
+14.685 
+ 2.5 

-13. 296 
-13.3 

+ 5.825 
+ 5.8 

+ 3.252 
+ 3.2 

+86.021 
-10. 753 

+ 7.336 
- 1.223 

15 

16 

17 

18 

23 

19 

20 

21 

+ 1.282 
-43. 806 
+44. 790 

- 3.156 

- 2.824 

- 4.549 
+ 14. 685 
+44. 790 
-52.6 

- 0.843 

- 3.389 

- 3.956 
+14.347 
+44. 790 
-54.8 

- 2.620 
+25.267 
+44. 790 
-62. 2 

+ 5.237 
+ 5.3 

- 0.843 

- 3.889 

- 3.156 

- 2.824 

- 9.843 
+ 10.493 
-169.6 

-179.162 
+ 44.790 

- 0.480 

- 1.845 
+ 4.3 

- 1.514 
+ 2.541 
+ 6.4 

- 0.507 
+ 2.918 
+ 1.1 

+ 2.266 
+ 2.2 

+ 1.975 
+ 2.0 

+ 7.427 
+ 7.5 

+ 3.511 
+ 3.5 

- 3.654 

- 3.7 

- 3.851 

- 3.9 

22 

23 

24 


+ 0.125 
+ 3.097 
- 1.1 

+ 0.588 
+ 3.593 
+ 2. 5 

+ 2.007 
+ 7.377 
- 1.9 

+ 2.122 
+ 2.1 

+ 6.681 
+ 6.7 

+ 7.484 
+ 7.5 














































































130 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 25 . 


Final computation of triangles 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

1 1- 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 

-1+ 3 
+24 
-22 

Duck-Larrabee 

Gunner 

Duck 

Larrabee 

O t tt 

45 43 29.4 
124 02 00.9 
10 14 25.7 

tt 

- 1.4 
+ 7.5 

- 2.1 

tt 

28.0 

08.4 

23.6 


O t tt 

3.410111 

0.145093 

9.918392 

9.249858 


Gunner-Larrabee 

Gunner-Duck 


+ 4.0 




3. 473596-1 
2.805062 

-1+ 4 
+24 
-21 

Duck-Mam 

Gunner 

Duck 

Mam 

108 41 53.9 
57 00 57.8 
14 17 07.7 

- 3.4 
+ 7.5 

- 3.5 

50.5 

65.3 

04.2 



3.389282 

0.023547 

9.923681 

9.392234 


Gunner-Mam 

Gunner-Duck 


+ 0.6 

* 



3.336510 

2.805063-1 

-1+ 5 
+24 

Duck-Lubec Channel Light¬ 
house 

Gunner 

Duck 

Lubec Channel Lighthouse 
Gunner-Lubec Channel 
Lighthouse 

Gunner-Duck 

115 34 42.9 
33 11 41.9 
35.2 

+ 4.0 
+ 7.5 

46.9 

49.4 

23.7 

l 

31 13 23.7 

3.045619 

0.044800 

9.738400 

9. 714643 

2.828819 

2.805062 

-2+ 3 
+23 
-22 

Indian Point-Larrabee 

Gunner 

Indian Point 

Larrabee 

31 57 41.4 
114 00 36.0 
34 01 32.0 

+ 6.0 
+ 6.7 
- 2.1 

47.4 

42.7 

29.9 



3.236668 

0.276237 

9.960690 

9. 747842 

• 

Gunner-Larrabee 
Gunner-Indian Point 


+10.6 




3.473595 

3.260747 

-2+ 4 
+23 
-21 

Indian Point-Mam 

Gunner 

Indian Point 

Mam 

94 56 05.9 
47 05 36.3 
37 58 10.6 

+ 4.0 
+ 6.7 
- 3.5 

09.9 

43.0 

07.1 



3.470097 

0.001614 

9.864800 

9. 789037 


Gunner-Mam 

Gunner-Indian Point 

i 


+ 7.2- 




3.336511-1 

3.260748-1 

-2+ 5 
+23 

Indian Point-Lubec Channel 
Lighthouse 

Gunner 

Indian Point 

Lubec Channel Lighthouse 
Gunner-Lubec Channel 
Lighthouse 

Gunner-Indian Point 

101 48 54.9 
18 35 56.3 
68.8 

+11.4 
+ 6.7 

66.3 

63.0 

50.7 


59 34 50.7 

3.315762 

0.009305 
9.503754 

9.935680 

2.828821-1 

3.260747 


i 

i. 



\ 


























APPLICATION OF LEAST SQUARES TO TRIANGULATION. 131 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 

- 3 + 4 
+22 

-21 

Larrabee-Mam 

Gunner 

Larrabee 

Mam 

o t ft 

62 58 24.5 
44 10 37.0 
72 50 61.9 

n 

- 2.0 
+ 2.1 
- 3.5 

// 

22.5 

39.1 

58.4 


O lit 

3.443126 

0.050224 

9.843160 

9.980246 


Gunner-Mam 

Gunner-Larrabee 


- 3.4 




3.336510 

3 . 473596-1 

- 3 + 5 
+22 

Larrabee-Lubec Channel 
Lighthouse 

Gunner 

Larrabee 

Lubec Channel Lighthouse 
Gunner-Lubec Channel 
Lighthouse 
Gunner-Larrabee 

69 51 13.5 
12 59 29.9 
16.6 

+ 5.4 
+ 2.1 

18.9 

32.0 

09.1 


97 09 09 . 1 

3.449573 

0.027415 

9.351833 

9.996607 
2 . 828821-1 

3.473595 

- 3 + 7 
+22 

Larrabee-Lubec church spire 
Gunner 

Larrabee 

Lubec church spire 
Gunner-Lubec church spire 
Gunner-Larrabee 

154 32 48.4 
10 46 00.0 
71.6 

+ 13.8 
+ 2.1 

62.2 

02.1 

55.7 


14 40 55.7 

3.702872 

0.366819 

9.271423 

9.403903 
3 . 341114+1 
3 . 473594+1 

- 4 + 5 
+21 

Mam-Lubec Channel Light¬ 
house 

Gunner 

Mam 

Lubec Channel Lighthouse 
Gunner-Lubec Channel Light¬ 
house 

Gunner-Mam 

6 52 49.0 

3 04 49.6 
21.4 

+ 7.4 
+ 3.5 

56.4 
53.1 

10.5 


170 02 10.5 

3 . 176968 

0.921432 

8 . 730418 

9.238109 
2 . 828818+2 

3 . 336509+1 

+ 2 - 7 

-23 

Lubec church spire-Indian 
Point 

Gunner 

Lubec church spire 

Indian Point 

Gunner-Indian Point 
Gunner-Lubec church spire 

173 29 30.2 
56 55.4 

3 33 34.4 

- 19.8 

- 6.7 

10.4 

81.9 

27.7 


2 57 21.9 

3.603122 

0.945226 

8.712401 

8.792767 

3 . 260749-2 

3.341115 

- 9+10 
- 5 + 8 

Gunner-Lubec Channel Light¬ 
house 

Cranberry Point 

Gunner 

Lubec Channel Lighthouse 
Cranberry Point-Lubec Chan¬ 
nel Lighthouse 

Cranberry Point-Gunner 

75 45 12.8 
90 05 58 . 7 
48.5 

+ 1.4 
- 3.7 

14.2 

55.0 

50.8 


14 08 50.8 

2.828820 

0.013565 

9.999999 

9.388133 

2.842384 

2.230518 




















132 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 23 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

l 

Plane 

angle 

Loga¬ 

rithm 



Of It 

ft 

It 


O t It 



Gunner-Mam 






3.336510 

-9+11 

Cranberry Point 

78 36 65.9 

-14.6 

51.3 



0.008632 

-4+ 8 

Gunner 

96 58 47. 7 

+ 3.7 

51.4 



9.996708 

-20+21 

Mam 

4 24 21.3 

- 4.0 

17.3 



8.885376 




-14.9 






Cranberry Point-Mam 






3.341910 


Cranberry Point-Gunner 






2.230518 


Gunner-Lubec church spire 






3.341115 

-9+13 

Cranberry Point 

174 08 32.3 

+ 4.5 

36.8 



0.991245 

-7+ 8 

Gunner 

5 24 23.8 

-12.1 

11.7 



8.973889 


Lubec church spire 

03.9 


11.5 


0 27 11.5 

7.898156 


Cranberry Point-Lubec 






3.306249 


church spire 








Cranberry Point-Gunner 






3.230516+ 2 


Lubec Channel Lighthouse- 






3.176968 


Mam 







-10+11 

Cranberry Point 

2 51 53.1 

-16.0 

37.1 



1.301891 


Lubec Channel Lighthouse 

35.2 


58.7 


175 48 58. 7 

8.863051 

-20 

Mam 

1 19 31.7 

- 7.5 

24.2 



8.363526 


Cranberry Point-Mam 






3.241910 


Cranberry Point-Lubec Chan- 






2.842385*1 


nel Lighthouse 








Lubec church spire-Treat 2 






3.309982 

-13+14 

Cranberry Point 

17 44 02.9 

- 2.6 

00.3 



0.516286 


Lubec church spire 

19.0 


23.6 


144 41 23.6 

9.761929 

-19 

Treat 2 

17 34 38.1 

- 2.0 

36.1 



9.479981 


Cranberry Point-Treat 2 






3.588197 


Cranberry Point-Lubec 






3.306249 


church spire 








Lubec church spire-Cran- 






3.306249 


berry Point 







-15+16 

Telegraph 

128 38 59.9 

- 5.9 

54.0 



0.107352 


Lubec church spire 

47.5 


47.8 


30 51 47. 8 

9.710110 

-12+13 

Cranberry Point 

20 29 12.6 

+ 5.6 

18.2 



9.544090 


Telegraph-Cranberry Point 






3.123711 


Telegraph-Lubec church spire 






2.957691 


Lubec church spire-Gunner 






3.341115 

-15+17 

Telegraph 

131 33 53.1 

- 6.1 

47.0 



0.125967 


Lubec church spire 

36.1 


36.3 


30 24 36.3 

9.704310 

-6+ 7 

Gunner 

18 01 30. 8 

+ 5.9 

36.7 



9. 490609 


Telegraph- Gunner 






3.171392 


Telegraph-Lubec church spire 






2.957691 


Cranberry Point-Gunner 






2. 230518 

-16+17 

Telegraph 

2 54 53. 2 

- 0.2 

53.0 



1.293713 

-9+12 

Cranberry Point 

153 39 19. 7 

- 1.1 

18.6 



9. 647161 

-6+ 8 

Gunner 

23 25 54. 6 

- 6.2 

48.4 



9.599479 




- 7.5 






Telegraph-Gunner 






3.171392 


Telegraph-Cranberry Point 






3.123710+1 

















APPLICATION OF LEAST SQUARES TO TRIANGULATION 


133 


Final computation of triangles —Continued 


Symbol 

Station. 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 



o / // 

// 

// 


Of ff 



Cranberry Point-Lubec Chan- 






2. 842384 


nel Lighthouse 







-16+18 

Telegraph 

29 52 44.5 

+ 9.0 

53.5 



0.302589 

-10+12 

Cranberry Point 

77 54 06.9 

- 2.5 

04.4 



9. 990245 


Lubec Channel Lighthouse 

08.6 


02.1 


72 13 02.1 

9. 978738 


Telegranh-Lubec Channel 






3.135218 


Lighthouse 

Telegraph-Cranberry Point 






3.123711 


V 

Gunner-L u b e c Channel 






2.828820 


Lighthouse 






0.343447 

-17+18 

Telegraph 

26 57 51.3 

+ 9.2 

60.5 



-5+6 

Gunner 

65 40 04.1 

+ 2.5 

06.6 



9.962951 


Lubec Channel Lighthouse 

64.6 


52.9 


86 21 52.9 

9.999125 


Telegraph-Lubec Channel 
Lighthouse 






3.135218 


Telegraph-Gunner 






3.171392 























134 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation, 

STATION GUNNER 














O 

/ 

ft 

a 

Duck to Lubec Channel Lighthouse 




86 

26 

42.1 

Second angle 

Lubec Channel Lighthouse and Gunner 



+33 

11 

49.4 

a 

Duck to Gunner 







119 

38 

31.5 

Act 












— 


17.8 













180 



«' 

Gunner to Duck 






. 

299 

38 

13.7 








First angle o 

f triangle 

115 

34 

46.9 

4> 

O 

44 



/ 

50 

ft 

33. 886 


Duck 


X 

66 

57 

47.822 

A<j> 

+ 




10.227 





A\ 

+ 


25. 263 

<t>' 

44 



50 

44.113 

Gunner 


X' 

66 

58 

13.085 


O / 

n 



s 

2. 805062 


S 2 


5.6101 




44 50 

39 

COS a 

9.694237 


sin 2 a 


9.8782 




ft 




B 

8.510480 


C 


1.4016 



1st term 

-10. 2277 


h 

1.009779 




6.8899 



2d term 

+ 0.0008 












— A<J> 

-10. 2269 













s 



2. 805Q62 











sin a 



9.939086 











A' 



8.508994 

A\ 


1. 402490 





sec <£' 



0.149348 

sin U<t>+<t>') 

9.848301 








1. 402490 




1. 250791 









r 





ft 






JX 



+25.2633 


Act 


+17.82 





STATION CRANBERRY POINT 








O 

/ 

ft 

a 

Gunner to Lubec Channel Lighthouse 


55 

13 

00.6 

Second angle 

Lubec Channel Lighthouse and Cranberry Point 

+ 90 

05 

55.0 

a 

Gunner to Cranberry Point 



145 

18 

55.6 

Act 






— 


03.1 







180 



a' 

Cranberry Point to Gunner 



325 

18 

52.5 





First angle of triangle 

75 

45 

14.2 


O 

r 

ft 






<t> 

44 

50 

44.113 

G turner 

X 

66 

58 

13.085 

A<f> 

+ 


04.529 


A\ 

+ 


04. 406 

V 

44 

50 

48.642 

Cranberry Point 

X' 

66 

58 

17. 491 



o r rr 

s 

2.230518 

s 2 

4.4610 


44 50 46 

COS a 

9.915029 

sin 2 a 

9.5103 


ft 


B 

8.510480 

C 

1. 4016 

1st term 

-4.5293 

h 

0.656027 


5.3729 

2d term 

+0.0000 





-A4> 

• -4.5293 






S 


2.230518 





sin a 


9. 755156 





A' 


8.508994 

A\ 

0.644025 


sec <f>’ 


0.149357 

sin *(<£+<£') 

9.848315 




0.644025 


0. 492340 




ft 



ft 


A\ 


+4.4058 

—Aa 

+3.1 

























































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 135 


secondary triangulation 


STATION GUNNER 


Third angle 

CL 

Joe 


* 

A<t> 

V 

£($+<£') 


1st term 
2d term 

— d<j> 


Lubec Channel Lighthouse to Duck 
Gunner and Duck 

Lubec Channel Lighthouse to Gunner 


Gunner to Lubec Channel Lighthouse 


44 


+ 


50 


44 


44 50 36 


-12. 4620 
+ 0.0008 


50 


31.652 
12.461 


44.113 


Lubec Channel 
Lighthouse 


-12.4612 
s 

sin a 
A' 

sec <£' 


JX 


s 

COS a 
B 


2.828819 
9.914485 
8.508994 
0.149348 


2.828819 
9.756288 
8.510480 


Gunner 
«2 

sin2 a 

C 


1. 401646 


-25. 2142 


1.095587 


JX 

sin 


—Joe 


X 

JX 

X' 


o 

266 
- 31 

/ 

26 

13 

it 

06.5 

23.7 

235 

12 

42.8 

+ 


17.8 

180 

55 

13 

00.6 

66 

58 

38. 299 

- 


25.214 

66 

58 

13.085 


5.6576 
9.8290 
1. 4016 


6.8882 


1.401646 
9.848294 


1.249940 


-17.78 


STATION CRANBERRY POINT 








o 

/ 

n 

a 

Lubec Channel Lighthouse to G unner 


235 

12 

42.8 

Third angle 

Cranberry Point and Gunner 


- 14 

08 

50.8 

CL 

Lubec Channel Lighthouse to Cranberry Point 

221 

03 

52.0 

da 






+ 


14.7 







180 



a' 

Cranberry Point to Lubec Channel Lighthouse 

41 

04 

06.7 

ef> 

O 

44 

t 

50 

it 

31.652 

Lubec Channel 

X 

66 

.58 

38.299 





Lighthouse 





deft 

+ 


16.990 


JX 

— 


20.808 

<t>' 

44 

50 

48.642 

Cranberry Point 

X' 

66 

58 

17.491 



O / ft 

s 

2.842384 

S2 

5.6848 

§($+<£') 

44 50 40 

COS a 

9.877355 

sin 2 a 

9.6350 

It 

B 

8.510480 

C 

1.4016 

1st term 

-16.9910 

h 

1.230219 


6.7214 

2d term 

+ 0.0005 





— d<f> 

-16.9905 






8 

sin a 
A' 

sec <f>' 


JX 


2.842384 
9.817504 
8.508994 
0.149357 


1.318239 

tt 

-20.8084 


JX 

sin £($+<£') 


—Joe 


1.318239 
9.848303 


1.166542 

it 

-14,67 


























































136 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28 


Final position computation , 


STATION TELEGRAPH 


/ 


o 

f 

ft 

a 

01 

06.7 

+ 77 

51 

04.4 

118 

58 

11.1 

— 


37.4 

180 



298 

57 

33.7 

29 

52 

53.4 

C6 

58 

17. 491 

+ 


52. 973 

66 

59 

10. 464 


Second angle 


a 

Act 


j<i> 


Cranberry Point to Lubec Channel Lighthouse 
Lubec Channel Lighthouse and Telegraph 

Cranberry Point to Telegraph 


Telegraph to Cranberry Point 


44 


+ 


44 


50 


48. 642 
20. 858 


51 


09.500 
+ 1 


First angle of triangle 
Cranberry Point 
Telegraph 


s 

sin a 
A’ 

sec <£' 


JX 


3.123711 
9.941946 
8 . 508994 
0.149401 


1. 724052 


+52.9728 


sin ^(<£+ 9 /) 


—da 


X 

JX 



o t n 

s 

3.123711 

$2 

6 . 2474 

£(<£+<£') 

44 50 59 

COS a 

9.685157 

sin 2 a 

9. 8839 

tt 

B 

8 . 510480 

C 

1. 4016 

1 st term 

-20. 8617 

h 

1.319348 


7.5329 

2 d term 

+ 0.0034 





—J<j> 

-20. 8583 






1. 724052 
9. 848343 


1.572395 


+37.36 































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 137 


secondcinj triangulation —Continued. 

STATION TELEGRAPH 


Third angle 

a 

Act 


J(J} x 


Lubec Channel Lighthouse to Cranberry Point 
Telegraph and Cranberry Point 

Lubec Channel Lighthouse to Telegraph 


Telegraph to Lubec Channel Lighthouse 


44 


+ 


44 


r>o 


51 


31. 652 
37.849 


09. 501 


Lubec Channel 
Lighthouse 


Telegraph 


$ 

sin a 
A’ 

sec 4>' 


JX 


3.135218 
9. 713762 
8. 508994 
0.149401 


1.507375 
// 

+32.1644 


JX 

sin 


—Ja 


X 

JX 

X' 


o 

221 
- 72 

/ 

03 

13 

n 

52.0 

02.2 

148 

50 

49.8 

22.7 

180 

328 

50 

27.1 

66 

58 

38.299 

+ 


32.164 

66 

59 

10. 463 
+1 



o r n 

s 

3.135218 

«2 

6.2704 


44 50 50 

COS a 

9.932367 

sin 2 a 

9. 4275 


// 

B 

8. 510480 

C 

1.4016 

1st term 

-37. 8500 

h 

1. 578065 


7.0995 

2d term 

+ 0.0012 





— J4> 

-37.8488 






1. 507375 
9. 848324 


1.355699 

n 

+ 22.69 





























138 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Computation of probable errors 



Adopted 

v’s 

first 

solution 

. v 2 


Adopted 

v r s 

second 

solution 

V i 

1 

+ 0.2 

0.04 

1 

- 1.4 

1.96 

2 

- 8.5 

72.25 

2 

- 8.8 

77.44 

3 

- 2.7 

7.29 

3 

- 2.8 

7.84 

4 

- 4.9 

24.01 

4 

- 4.8 

23.04 

5 

+ 3.1 

9.61 

5 

+ 2.6 

6.76 

6 

+ 4.9 

24.01 

6 

+ 5.1 

26.01 

7 

+ 9.5 

90.25 

7 

+ 11.0 

121.00 

8 

-1.5 

2.25 

8 

- 1.1 

1.21 

9 

+ 1.7 

2.89 

9 

+ 1.3 

1.69 

10 

+ 3.1 

9.61 

10 

+ 2.7 

7.29 

11 

-13.1 

171.61 

11 

-13.3 

176.89 

12 

+ 0.6 

0.36 

12 

+ 0.2 

0.04 

13 

+ 5.0 

25.00 

13 

+ 5.8 

33.64 

14 

+ 2.9 

8.41 

14 

+ 3.2 

10.24 

15 

+ 2.3 

5.29 

15 

+ 2.2 

4.84 

16 

- 3.8 

14.44 

16 

- 3.7 

13.69 

17 

- 3.8 

14. 44 

17 

- 3.9 

15.21 

18 

+ 5.4 

29.16 

18 

+ 5.3 

28.09 

19 

+ 0.8 

0.64 

19 

+ 2.0 

4/00 

19' 

- 0.7 

0.49 

20 

+ 7.5 

56.25 

20 

+ 5.6 

31.36 

21 

+ 3.5 

12.25 

21 

+ 2.1 

4.41 

22 

+ 2.1 

4.41 

20' 

- 0.8 

0.64 

23 

+ 6.7 

44.89 

20" 

- 1.5 

2.25 

24 

+ 7.5 

56.25 

20'" 

- 6.5 

42.25 




20"" 

+ 1.2 

1.44 



734.93 

22 

+ 2.6 

6.76 




22' 

- 0.7 

0.49 




22" 

- 3.7 

13.69 




22"' 

+ 5.7 

32.49 




22"" 

- 0.6 

0.36 




22 v 

- 3.3 

10.89 




23 

+ 1.9 

3.61 




23' 

- 4.2 

17.64 




23" 

- 5.5 

30.25 




23'" 

- 2.8 

7.84 




23"" 

+10.7 

114.49 




24 

+ 6.5 

42.25 




24' 

- 3.0 

9.00 




24" 

- 3.4 

11.56 






895. 72 





In the first method of adjustment, there are 40 equations to deter¬ 
mine 14 unknown quantities, namely, 6 ^ 0 ’s and dl’s and 82 ’s. The 
probable error of an observed direction is therefore, 

°- 6745 Vw= ±4: ° 

In the second method there are 24 equations to determine 9 
unknown quantities, namely, 6<50’s and £A’s and 3z ’s at new points. 
The probable error of an observed direction is therefore, 

„. 67 „^ig3. i4 : 7 
















APPLICATION OF LEAST SQJJARES TO TRIANGULATION. 139 


ADJUSTMENT OF A FIGURE CONTAINING LATITUDE, LONGITUDE, 
AZIMUTH, AND LENGTH CONDITIONS BY THE METHOD OF VARIA¬ 
TION OF GEOGRAPHIC COORDINATES 

This example illustrates the fitting of a chain of triangulation in 
between fixed lines at the ends. The necessary preliminary computa¬ 
tions of the assumed positions and directions could have been carried 
out in much the same way as in the preceding examples. A prelimi¬ 
nary figure adjustment was, however, available and the results of it 
were used in the preliminary computations of the triangles and the 
geographic positions, pages 140-157. The fixed lines at the ends of the 
chain are shown in the following list of fixed positions. The list of 
observed directions is not given. The necessary data may be derived 
from the observed angles of the triangles, pages 140-143, taken in 
connection with figure 7, page 157. 


Table of fixed positions 


Station 

Latitude 

and 

longitude 

Azimuth 

Back 

azimuth 

To station 

Loga¬ 
rithm of 
distance. 


O / II 

o in 

o / // 



Fort Morgan. 

30 13 42.242 
88 01 23.228 





Dauphin Island east base. 

30 14 56.379 
88 08 14.288 

281 42 17.9 

101 45 44.9 

Fort Morgan. 

4.050203 

Dauphin Island west base. 

30 14 21.492 
88 14 51.034 

264 11 22.1 

84 14 41.9 

Dauphin Island east base 

4.027832 

Biloxi Lighthouse. 

30 23 39.419 
88 54 03.820 





Ship Island Lighthouse. 

30 12 45.341 
88 57 57.464 

197 12 19.7 

17 14 17.6 

Biloxi Lighthouse. 

4.323998 











140 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary computation of triangles 


Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 


o / n 

tf 

// 

// 

Of// 


Fort Morgan-Dauphin Island east base 






4. 050203 

Cedar 

43 5-4 21.8 

+ 0.0 

21.8 

0.1 

21.7 

0.158968 

Fort Morgan 

42 33 37. 7 

+ 0.3 

38.0 

0.1 

37.9 

9.830183 

Dauphin Island east base 

93 31 59. 6 

+ 0.9 

60.5 

0.1 

32 00. 4 

9.999174 



+ 1.2 


0.3 



Cedar-Dauphin Island east base 






4.039354 

Cedar-Fort Morgan 






4.208345 

Dauphin Island east base-Dauphin Is- 






4. 027832 

land west base 







Cedar 

37 26 33.0 

- 0.9 

32.1 

0.1 

32.0 

0.216124 

Dauphin Island east base 

103 55 36. 4 

- 0.9 

35.5 

0.1 

35.4 

9.987043 

Dauphin Island west base 

38 37 52.5 

+ 0.2 

52.7 

0.1 

52.6 

9. 795398 



- 1.6 


0.3 



Cedar-Dauphin Island west base 






4.230999 

Cedar-Dauphin Island east base 






4.039354 

Cedar-Dauphin Island east base 






4.039354 

Cat 

69 30 32. 7 

+ 0.8 

33.5 

0.0 

33.5 

0.028386 

Cedar 

60 11 14.2 

+ 1-4 

15.6 

0.1 

15.5 

9.938349 

Dauphin Island east base 

50 18 11.4 

- 0.3 

11.1 

0.1 

11.0 

9.886171 



+ 1.9 


0.2 



Cat-Dauphin Island east base 






4.006089 

. Cat-Cedar 






3.953911 

Cedar-Dauphin Island west base 






4.230999 

Cat 

135 31 54.9 

+ 2.7 

57.6 

0.0 

57.6 

0.154590 

Cedar 

22 44 41.2 

+ 2.3 

43.5 

0.1 

43.4 

9. 587303 

Dauphin Island west base 

21 43 18. 7 

+ 0.4 

19.1 

0.1 

19.0 

9. 568322 



+ 5.4 


0.2 



Cat-Dauphin Island west base 






3.972892 

Cat-Cedar 






3.953911 

Dauphin Island east base-Dauphin Is- 






4. 027832 

land west base 







Cat 

66 01 22.2 

+ 1.9 

24.1 

0.1 

24.0 

0.039191 

Dauphin Island east base 

53 37 25.0 

- 0.6 

24.4 

0.1 

24.3 

9.905869 

Dauphin Island west base 

60 21 11.2 

+ 0.6 

11.8 

0.1 

11.7 

9. 939066 



+ 1.9 


0.3 



Cat-Dauphin Island west base 






3.972892 

Cat-Dauphin Island east base 

. 





4.006089 

Cedar-Cat 






3.953911 

Pins 

23 22 40.3 

- 1.8 

38.5 

0.1 

38.4 

0. 401445 

Cedar 

30 54 61. 5 

- 3.5 

58.0 

0.1 

57.9 

9. 710779 

Cat 

18.4 


23.7 

0.0 

125 42 23. 7 

9.909565 





0.2 



Pins-Cat 






4.066135-1 

Pins-Cedar 






4.264921 

Cedar-Dauphin Island west base 






4.230999 

Pins 

58 45 26.0 

- 2.8 

23.2 

0.2 

23.0 

0.068049 

Cedar 

53 39 42. 7 

- 1.2 

41.5 

0.2 

41.3 

9.906082 

Dauphin Island west base 

67 34 57.9 

- 2.0 

55.9 

0.2 

55.7 

9.965873 



- 6.0 


0.6 



Pins-Dauphin Island west base 






4.205130 

Pins-Cedar 






4.264921 

Cat-Dauphin Island west base 






3.972892 

Pins 

35 22 45. 7 

- 1.0 

44.7 

0.1 

44.6 

0.237334 

Cat 

35.3 


38.7 

0.0 

98 45 38. 7 

9.994903 

Dauphin Island west base 

45 51 39.2 

- 2.4 

36.8 

0.1 

36.7 

9.855908 





0.2 



Pins-Dauphin Island west base 






4.205129+1 

Pins-Cat 






4.066134 































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 141 


Preliminary computation of triangles —Continued 


Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 


O / // 

n 

// 

n 

o / n 


Pins-Dauphin Island west base 






4.205130 

Grand 

54 52 01. 6 

+ 1.4 

00.2 

0.2 

00.0 

0.087345 

Pins 

85 13 07. 0 

+ 1.3 

08.3 

0.1 

08.2 

9. 998486 

Dauphin Island west base 

39 54 50.9 

+ 1.1 

52.0 

0.2 

51.8 

9.807293 



+ 1.0 


0.5 



Grand-Dauphin Island west base 






4. 290961 

Grand-Pins 






4.099768 

Grand-Pins 






4.099768 

Petit 

33 09 OS. 7 

- 0.3 

08.4 

0.1 

08.3 

0. 262119 

Grand 

114 03 53.6 

- 0.6 

53.0 

0.1 

52.9 

9.960511 

Pins 

32 46 57.5 

+ 1.4 

58.9 

0.1 

58.8 

9. 733565 



+ 0.5 


0.3 



Petit-Pins 






4.322398 

Petit-Grand 






4.095452 

Grand-Dauphin Island west base 






4. 290961 

Petit 

81 41 28.2 

+ 0.1 

28.3 

0.1 

28.2 

0.004583 

Grand 

59 11 52.0 

+ 0.8 

52.8 

0.2 

52.6 

9. 933964 

Dauphin Island west base 

39 06 39.1 

+ 0.3 

39.4 

0.2 

39.2 

9.799908 



+ 1.2 


0.5 



Petit-Dauphin Island west base 






4.229508 

Petit-Grand 






4. 095452 

Pins-Dauphin Island west base 




0.2 

19.7 

4.205130 

Petit 

48 32 19. 5 

+ 0.4 

19.9 

0.125284 

Pins 

52 26 09. 5 

- 0.1 

09.4 

0.2 

09.2 

9.899093 

Dauphin Island west base 

79 01 30. 0 

+ 1.4 

31.4 

0.3 

31.1 

9. 991984 



+ 1.7 


0.7 


4.229507+ 1 

Petit-Dauphin Island west base 






Petit-Pins 






4.322398 

Grand-Petit 





23.5 

4.095452 

Pascagoula 

37 39 20. 6 

+ 3.0 

23.6 

0.1 

0. 214011 

Grand 

104 18 56.1 

+ 1.5 

57.6 

0.2 

57.4 

9. 986300 

Petit 

38 01 38. 6 

+ 0.6 

39.2 

0.1 

39.1 

9. 789609 



+ 5.1 


0.4 


4.295763 

Pascagoula-Petit 






Pascagoula-Grand 






4.099072 

Pascagoula-Grand 




0.1 

42.7 

4.099072 

Horn 

38 49 39. 0 

+ 3.8 

42.8 

0.202738 

Pascagoula 

97 55 54. 0 

+ 3.5 

57.5 

0.2 

57.3 

9.995824 

Grand 

43 14 18. 9 

+ 1.2 

20.1 

0.1 

20.0 

9.835717 



+ 8.5 


0.4 


4. 297634+1 

Horn-Grand 





H om- Pascagoula 






4.137527 

Pascagoula-Petit 

Horn 

77 06 13.2 

+ 2.9 

16.1 

0.2 

15.9 

4. 295763 

0. 011094 

Pascagoula 

Petit 

60 16 33. 4 

+ 0.5 

33.9 

0.2 

33.7 

9.938732 

42 67 10. 3 

+ 0.3 

10.6 

0.2 

10.4 

9. 830670 



+ 3.7 


0.6 



Horn-Petit 





4. 245589 

H orn-Pascagoula 






4.137527 

Grand-Petit 

Horn 

38 16 34. 2 

- 0.9 

33.3 

0.2 

33.1 

4. 095452 

0.207995 

Grand 

61 04 37. 2 

+ 0.3 

37.5 

0.2 

37.3 

9.942142 

Petit 

80 38 48. 9 

+ 0.9 

49.8 

0.2 

49.6 

9. 994188 



+ 0.3 


0.6 



Horn-Petit 

Horn-Grand 





4.245589 

4. 297635 









































142 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Preliminary computation of triangles —Continued 


Station 

Observed 

angle 

Correc¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane 

angle 

Loga¬ 

rithm 



Off/ 

ff 

ff 

// 

Of ff 


Pascagoula-Horn 

Belle 


48 58 49. 8 

- 1.8 

48.0 

0.2 

47.8 

4.137527 

0.123352 

Pascagoula 

Horn 


69 43 28. 0 
61 17 53. 5 

- 3.4 

- 5.6 

24.6 

47.9 

0.1 

0.2 

24.5 

47.7 

9.972217 

9.943058 




-10.8 


0.5 



Belle-Horn 







4.232096 

Belle-Pascagoula 







4.202937 

B elle-Pascagoula 





0.1 

44.3 

4. 202937 

Club 


58 40 43. 3 

+ 1.1 

44.4 

0.068406 

Belle 


89 28 55. 3 

- 0.6 

54.7 

0.2 

54.5 

9.999982 

Pascagoula 


31 50 23. 4 

- 2.1 

21.3 

0.1 

21.2 

9.722253 




- 1.6 


0.4 


4.271325 

Club-Pascagoula 







Club-Belle 







3.993596 

Belle-Horn 






56.4 

4.232096 

Club 


105 43 56.9 

- 0.4 

56.5 

0.1 

0.016582 

Belle 


40 30 05. 5 

+ 1.2 

06.7 

0.1 

06.6 

9. 812561 

Horn 


33 45 53.7 

+ 3.4 

57.1 

0.1 

57.0 

9.744918 




+ 4.2 


0.3 



Club-Horn 






4.061239 

Club-Belle 







3.993596 

Pascagoula-H orn 





0.1 

12.0 

4.137527 

Club 


47 03 13. 6 

- 1.5 

12.1 

0.135496 

Pascagoula 


37 53 04. 6 

- 1.3 

03.3 

0.1 

03.2 

9.788216 

Horn 


95 03 47. 2 

- 2.2 

45.0 

0.2 

44.8 

9. 998302 




- 5.0 


0.4 



Club-Horn 







4.061239 

Club-Pascagoula 







4.271325 

Belle-Club 







3.993596 

Doer 


41 02 10.7 

+ 0.7 

11.4 

0.1 

11.3 

0.182739 

Belle 


102 35 18.5 

+ 1.5 

20.0 

0.0 

20.0 

9.9S9432 

Club 


36 22 26.8 

+ 2.0 

28.8 

0.1 

28.7 

9. 773101 




+ 4.2 


0.2 



Deer-Club 







4.165767 

Deer-Belle 







3.949436 

Deer-Belle 







3.949436 

Ship 


33 10 60.2 

- 1.8 

58.4 

0.1 

58.3 

0.261764 

Deer 


97 49 37.9 

- 1.8 

36.1 

0.1 

36.0 

9.995935 

Belle 


48 59 22.9 

+ 2.9 

25.8 

0.1 

25.7 

9.877717 




- 0.7 


0.3 



Ship-Belle 







4.207135+ 1 

Ship-Deer 







4.0SS917 

Deer-Club 







4.165767 

Shin 


70 52 35. 0 

- 0.5 

34.5 

0.2 

34.3 

0.024654 

Deer 


56 47 27.2 

- 2.5 

24.7 

0.1 

24.6 

9.922554 

Club 


52 20 03.5 

- 2.3 

01.2 

0.1 

01.1 

9.898496 




- 5.3 


0.4 



Ship-Club 







4.112975 

Ship-Deer 

• 






4.08S917 

Belle-Club 







3.993596 

Ship 


37 41 34.8 

+ 1.3 

36.1 

0.1 

36.0 

0.213650 

Belle 


53 35 55. 6 

- 1.4 

54.2 

0.1 

54.1 

9. 905729 

Club 


88 42 30.3 

- 0.3 

30.0 

0.1 

29.9 

9.999S90 




- 0.4 


0.3 



Ship-Club 







4.112975 

Ship-Belle 







4.207136 

































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 143 


Preliminary computation of triangles —Continued 


Station 

Observed 

Correc- 

Spher- 

Spher- 

Plane 

Loga- 

angle 

tion 

angle 

excess 

angle 

rithm 


o / // 

// 

n 

// 

Of ff 


Deer-Ship 






4.088917 

Biloxi Lighthouse 

48 11 17.4 

+ 3.6 

21.0 

0.1 

20.9 

0. 1276*40 

Deer 

% 30 31.2 

+ 2.3 

33.5 

0.1 

33.4 

9.997191 

Ship 

35 18 04.4 

+ L4 

05.8 

0.1 

05.7 

9.761838 

Biloxi Lighthouse-Ship 


+ 7.3 


0.3 







4.213748 

Biloxi Lighthouse-Deer 






3.978395 

Biloxi Lighthouse-Deer 






3.97S395 

Ship Island Lighthouse 

25 30 02.1 

+ 1.4 

03.5 

0.2 

03.3 

0.366001 

Biloxi Lighthouse 

81 55 36.0 

- 1.7 

34.3 

0.1 

34.2 

9.995674 

Deer 

72 34 26.4 

- 3.7 

22.7 

0.2 

22.5 

9.979593 



- 4.0 


0.5 



Ship Island Lighthouse-Deer 






4.340070 

Ship Island Lighthouse-Biloxi Light¬ 
house 






4.323989 

Biloxi Lighthouse-Ship 






4.213748 

Ship Island Lighthouse 

50 31 41.2 

- 0.9 

40.3 

0.2 

40.1 

0.112420 

Biloxi Lighthouse 

33 44 18.6 

- 5.3 

13.3 

0.2 

13.1 

9. 744591 

Ship 

95 44 07.0 

- 0.1 

06.9 

0.1 

06.8 

9.997S21 



- 6.3 


0.5 



Ship Island Lighthouse-Ship 

Ship Island Lighthouse-Biloxi Light¬ 
house 






4.070759 

4.323989 

Deer-Ship 






4.088917 

Ship Island Lighthouse 

25 01 39.1 

- 2.3 

36.8 

0.1 

36.7 

0.373615 

Deer 

23 56 04.8 

+ 6.0 

10.8 

0.1 

10.7 

9.608227 

Ship 

131 02 11.4 

+ 1.3 

12.7 

0.1 

12.6 

9.887537 



+ 5.0 


0.3 



Ship Island Lighthouse-Ship 






4.070759 

Ship Island Lighthouse-Deer 






4. 340069+ 1 


91865°—15-10 




















144 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation , 


STATION CAT 


Second angle 

a. 

Joe 


Jej) 


1st term 
2d, 3d, and 1 
4th terms / 

-Jej, 


East base to west base 
West base and Cat 

East base to Cat 


Cat to east base 


First angle of triangle 


30 


+ 


14 

4 


30 

O / 

30 16 58 

It 

-244. 2407 
+ 0.0700 


19 


56.379 
04.171 


00.550 


Dauphin Island east 
base 


Cat 


X 

JX 

X' 


o 

t 

n 

84 

14 

41.9 

+53 

37 

24.4 

137 

52 

06. 3 

— 

2 

08.4 

180 

00 

00.00 

317 

49 

57.9 

66 

01 

24.1 

88 

08 

14.288 

+ 

4 

14.637 


88 


12 


28.925 


-244.1707 


S 

4.006089 

s 2 

8.0122 



-h 

2.388 

COS a 

9. 870173 

sin 2 a 

9. 6532 

h 2 

4. 776 

s 2 sin 2 a 

7.665 

B 

8. 511556 

C 

1.1712 

D 

2.332 

E 

5.917 

h 

2.387818 


8.8366 


7.108 


5.970 




+0.0686 


+0.0013 


+0.0001 


s 

4.006089 


sin a 

9. 826616 


A' 

8. 509352 

JX 

sec <f>' 

0.063865 

sin § 


2. 405922 


JX 

n 

+254. 6373 

—Act 


2. 405922 
9. 702663 


2.108585 

It 

+128.40 


STATION CEDAR 








o 

I 

II 

a 

East base to west base 



84 

14 

41.9 

Second angle 

West base and Cedar 



+103 

55 

35.5 

a 

East base to Cedar 



188 

10 

17.4 

Ace 






+ 


29.4 







180 

00 

00.00 

a' 

Cedar to east base 



8 

10 

46.8 





First angle of triangle 

37 

26 

32.1 


O 

I 

II 






</> 

30 

14 

56.379 

Dauphin Island east 

X 

88 

08 

14. 288 




base 





A<f> 

+ 

5 

51.941 


JX 

— 


58. 265 

V 

30 

20 

48.320 

Cedar 

X' 

88 

07 

16.023 



+ 1 







i(<£+<£') 


1st term 
2d and 3d \ 
terms / 

— A<f> 


O III 

s 

4.039354 

S 2 

8.0787 



30 17 52 

COS a 

9. 995568 

sin 2 « 

8.3054 

h 2 

5.093 

It 

B 

8.511556 

C 

1.1712 

D 

2.332 

-351.9476 

h 

2.546478 


7.5553 


7.425 

+ 0.0063 




+0.0036 


+0.0027 

-351.9413 








s 

sin a 
A' 

sec <£' 


JX 


4.039354 
9.152706 
8.509351 
0. 063997 


1. 765408 


-58.2650 


JX 

sin £(<£+<£') 


—jet 


1. 765408 
9. 702857 


1. 468265 


-29.39 







































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 145 


secondary triangulation 


STATION CAT 







. 

0 

/ 

// 

a 

West base to east baso 


. 

264 

11 

22.1 

Third angle 

Cat and east base 



- 60 

21 

11.8 

a 

West base to Cat 



203 

50 

10.3 

Joe 






+ 

1 

11.7 







180 

00 

00.00 

a ' 

Cat to west base 



23 

51 

22.0 


O 

t 

ft 







30 

14 

21.492 

Dauphin Island west 

X 

88 

14 

51.034 

A 




base 






+ 

4 

39.058 


JX 

— 

2 

22.109 

<t>’ 

30 

19 

00. 550 

Cat 

X' 

88 

12 

28.925 



O / // 

s 

3.972892 

«2 

7.9458 



-h 

2.445 

a(<£+<£') 

30 16 41 

COS a 

9. 961281 

sin 2 a 

9. 2130 

h 2 

4.891 

s 2 sin 2 a 

7.159 

ft 

B 

8. 511557 

C 

1.1711 

D 

2.331 

E 

5.917 

1st term 

-279.0808 

h 

2. 445730 


8.3299 


7. 222 


5.521 

2d,3d,and \ 
4th terms / 

+ 0.0231 




+0.0214 


+0.0017 



— J<t> 

-279.0577 










s 

sin a. 

A' 

sec <j>’ 

3.972892 
9.606514 
8.509352 
0.063865 

JX 

sin $(<£+<£') 

2.152623 
9. 702600 


2.152623 


1.855223 


n 


ft 

J\ 

-142.1095 

—Ja 

-71.65 


STATION CEDAR 












O 

/ 

ft 

cx 

West base to east base 





264 

11 

22.1 

Third angle 

Cedar and east base 






- 38 

37 

52.7 

CC 

West base to Cedar 






225 

33 

29.4 

Ja 










+ 

3 

49.5 











180 

00 

00.00 

a' 

Cedar to west base 






45 

37 

18.9 


O 


/ 


ft 








<t> 

30 


14 

21. 492 

Dauphin Island west 

X 

88 

14 

51.034 







base 





J<f> 

+ 


6 

26. 829 




JX 

— 

7 

35. 011 

4' 

30 


20 

48.321 


Cedar 

X' 

88 

07 

16.023 


O / // 

s 

4. 230999 

s 2 

8.4620 



-h 

2.588 

£(<£+<£') 

30 17 35 

COS a 

9. 845213 

sin 2 a 

9. 7073 

h 2 

5.176 

,$ 2 sin 2 a 

8.169 

ft 


B 

8. 511557 

C 

1.1711 

D 

2.331 

E 

5.917 

1st term 

-387.0517 

I 

L 

2.587769 


9.3404 


7. 507 


6.674 

2d,3d,and! 
4th terms / 

+ 0.2227 






+0.2190 


+0.0032 


+0.0005 

— J<1> 

-386.8290 












s 

sin a 
A' 

sec </>' 


JA 


4. 230999 
9. 853675 
8. 509351 

JX 

2.658022 

0.063997 

sin £(<£+<£') 

9. 702795 

2.658022 


2.360817 

ft 


. " 

—455.0111 

—Ja 

-229. 52 












































































146 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

Preliminary position computation , 


STATION PINS 





1 







O 

r 

/; 

a 

Cedar to west base* 






45 

37 

18.9 

Second angle 

West base and Pins 





+ 53 

39 

41.5 

a 

Cedar to Pins 







99 

17 

00.4 

Ja 










— 

5 

43.8 











180 

00 

00. 00 

a f 

Pins to 

Cedar 







279 

11 

16.6 








First angle of triangle 

58 

45 

23.2 


G 


9 


ft 








4> 

30 


20 

48.321 


Cedar 

X 

88 

07 

16.023 

J«f> 

+ 


1 

35.926 




JX 

+ 

11 

20. 238 


30 


22 

24. 247 


Pins 

X' 

■ 88 

18 

36. 261 












+1 


O / 

ft 

5 


4.264921 

s 2 

8.5298 



—h 

1.984 


30 21 

36 

COS a 

9. 207685 

sin 2 a 

9.9885 

h 2 

3.968 

s 2 sin 2 a 

8.518 

ft 


B 

8.511550 

C 

1.1729 

D 

2. 332 

E 

5. 919 

1st term 

-96. 4175 

h 

1.984156 


9. 6912 


6.300 


6.421 

2d,3d, andl 
4th terms / 

+ 0.4916 






+0. 4911 


+0.0002 


+0.0003 

-J<£ 

-95.9259 












s 

sin a 
A' 

sec <£' 


JX 


4. 264921 
9.994274 
8.509350 
0.064116 

JX 

sin £(<£+<£') 

2.832661 


ft 


+680.2382 

—Ja 


2.832661 
9. 703663 


2.536324 


+343.81 


STATION GRAND 








O 

t 

ft 

a 

Pins to west base 



337 

56 

39.8 

Second angle 

West base and Grand 



-I - 85 

13 

08.3 

a 

Pins to Grand 




63 

09 

48.1 

Ja 






— 

3 

32.3 







180 

00 

00.00 

a' 

Grand to Pins 




243 

06 

15.8 





First angle of triangle 

54 

52 

00.2 


O 

t 

ft 







30 

22 

24.247 

Pins 

X 

88 

18 

36. 262 

J(f> 

— 

3 

04. 658 


JX 

+ 

7 

00.260 

<y 

30 

19 

19. 589 

Grand 

X' 

88 

25 

36.522 


£(<£+(£') 


1st term 
2d,3d, andl 
4th terms / 

— J<J> 


30 20 52 

rt 

+ 184. 4693 
+ 0.1884 


+184.6577 
s 

sin a 
A' 

sec <f>' 


s 

4. 099768 

S 2 

8.1995 



-h 

2.266 

COS a 

9. 654608 

sin 2 a 

9.9010 

h 2 

4.532 

s 2 sin 2 a 

8.100 

B 

8.511548 

C 

1.1734 

D 

2.333 

E 

5.919 

h 

2. 265924 


9.2739 


6.865 


6.285 




+0.1879 


+0.0007 


-0.0002 


JX 


4.099768 
9.950510 
8.509352 

JX 

0. 063888 

sin 

2.623518 


// 

+ 420. 2600 

—Ja 


2.623518 
9. 703504 


2.327022 


+212.33 














































































0 


APPLICATION OP LEAST SQUARES TO TRIANGULATION. 147 


secondary triangulation —Continued 


STATION I’INS 








O 

/ 

tt 

a 

West base to Cedar 



225 

33 

29.4 

Third angle 

Pins and Cedar 




- 67 

34 

55.9 

a 

West base to Pins 



157 

58 

33.5 

Ja 






— 

1 

53.7 







180 

00 

00. 00 

a ' 

Pms to west base 



337 

56 

39.8 


O 

/ 

tt 







30 

14 

21. 492 

Dauphin Island west 

X 

88 

14 

51. 034 





base 





j<t> 

+ 

8 

02. 754 


JX 

+ 

3 

45. 228 


30 

22 

24. 246 

Pins 

X' 

88 

18 

36. 262 




+ 1 








O t tt 

30 18 23 

s 

COS a 

4. 205130 
9. 967092 

s 2 

sin 2 a 

8.4103 
9.1480 

h 2 

5.368 

-h 

s 2 sin 2 a 

2.684 

7. 558 


/./ 

B 

8. 511557 

C 

1.1711 

D 

2.331 

E 

5. 917 

1st term 

2d,3d,and 4 
4th. terms / 
-J <t> 

-482.8131 

+ 0.0587 

-482.7544 

h 

2. 683779 


8. 7294 

+0.0534 


7.699 

+0.0050 


6.159 

+0.0001 


s 

sin a 

A' 

sec <f>' 

4. 205130 
9. 574026 
8. 509350 
0. 064116 

JX 

sin $(<£+(£') 

2. 352622 
9. 702967 


2.352622 


2.055589 


ft 


tt 

JX 

+225.2278 

—Ja 

+ 113.66 


STATION GRAND 


West base to Pins 
Grand and Pins 

West base to Grand 


Grand to west base 



O 

/ 

tt 

* 

30 

14 

21. 492 

J(f> 

+ 

4 

58.097 


30 

19 

19. 589 



O t tt 

s 

4. 290961 

s 2 

8.5819 




30 16 50 

COS a 

9. 672485 

sin 2 a 

9. 8914 

h 2 

4.950 

tt 

B 

8. 511557 

C 

1.1711 

D 

2.331 

1st term 

-298. 5403 

h 

2.475003 


9.6444 


7.281 

2d, 3d, and \ 
4th terms / 

+ 0.4436 




+0. 4410 


+0.0019 

— d<f> 

-298.0967 








Dauphin Island west 
base 


Grand 


X 

JX 

X' 


O 

157 
- 39 

t 

58 

54 

tt 

33.5 

52.0 

118 

03 

41.5 

— 

5 

25.5 

180 

00 

00.00 

297 

58 

16.0 

88 

14 

51.034 

+ 

10 

45. 488 


88 


25 

-h 

s 2 sin 2 a 
E 


36.522 

2. 475 
8. 473 
5.917 


6.865 
+0.0007 


S 

sin a 

A' 

sec <f>' 

4.290861 
9.945687 
8. .509352 
0.063888 

JX 

sin !(<£+<£') 

2.809888 
9. 702633 


2.809888 


2. 512521 


tt 


tt 

JX 

+645. 4877 

—Ja 

+325. 48 











































































148 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Preliminary position computation, 

STATION PETIT 








O 

/ 

tt 

a 

Ci rand to west base 



297 

58 

16.0 

Second angle 

West base and Petit 



+ 59 

11 

52.8 

a 

Grand to Petit 




357 

10 

08.8 

A a 






+ 


11.6 







180 

00 

00.00 

a' 

Petit to Grand 




177 

10 

20.4 





First angle of triangle 

81 

41 

28.3 


O 

/ 

tt 






4> 

30 

19 

19.589 

Grand 

X 

88 

25 

36.522 

Ac}) 

— 

6 

44.089 


AX 

— 


23.005 

<}>' 

30 

12 

35.500 

Petit 

X' 

88 

25 

13.517 




1st term 
2d and 3d 1 
terms / 
— d<}> 


30 15 58 

ft 

+404.0853 
+ 0.0041 


+404.0894 
5 

sin « 

A' 

sec 4V 


dA 


s 

COS a 

B 


4.095452 
8.693623 
8.509354 
0.063392 


4.095452 
9.999470 
8.511551 


2. 606473 


sin 2 a 
C 


8.1909 
7.3872 
1.1725 


6.7506 
+0.0006 


h 2 

D 


5. 213 
2.332 


7. 545 
0.0035 


1.361821 

n 

-23.0049 


AX 

sin .}( 4>+4>') 


—Act 


1.361821 
9. 702445 


1.064266 

tt 

-11.59 


STATION HORN 








O 

/ 

ft 

a 

Grand to Petit 




357 

10 

08.8 

Second angle 

Petit and Horn 




+ 61 

04 

37.5 

a 

Grand to Horn 




58 

14 

46.3 

Act 






— 

5 

18.1 







180 

00 

00.00 

a ’ 

Horn to Grand 




238 

09 

28.2 





First angle of triangle 

38 

16 

33.’3 


O 

/ 

tt 






4> 

30 

19 

19.589 

Grand 

X 

88 

25 

36.522 

A<f> 

— 

5 

39.578 


AX 

+ 

10 

31.017 

</>’ 

30 

13 

40.011 

Horn 

A' 

88 

36 

07. 539 


i(4>+4> r ) 


1st term 
2d, 3d, andl 
4th terms / 
—Acj) 


O t tt 

s 

4.297635 

s 2 

8.5953 



-h 

2.530 

30 16 30 

COS a 

9. 721209 

sin 2 a 

9.8592 

h? 

5.061 

.S' 2 sin 2 a 

8.454 

tt 

B 

8.511551 

C 

1.1725 

D 

2.332 

E 

5.918 

+339.1525 

h 

2.530395 


9. 6270 


7.393 


6.902 

+ 0.4253 




+0. 4236 


+0.002o 


-0. 0008 

+339.5778 










s 

sin a 
A' 

sec <f>’ 


AX 


4.297635 
9.929581 
8.509354 
0.063471 

AX 

sin !(«£+<£') 

2.800041 


tt 


+631.0170 

—Act 


2. 800041 
9. 702560 


2.502601 


+318.13 


































































APPLICATION OF LEAST SQUARES TO TRIANGULATION, 149 


secondary triangulation —Continued 

STATION PETIT 








o 

/ 

ft 

cl 

West base to Grand 



118 

03 

41.5 

Third angle 

Petit and Grand 



- 39 

06 

39.4 

CL 

West base to Petit 



78 

57 

02.1 

Act 






— 

5 

13.4 







180 

00 

00.00 

a' 

Petit to west base 



258 

51 

48.7 

0 

o 

30 

t 

14 

ft 

21.492 

Dauphin Island west 

X 

88 

14 

51.034 

A<f> 

— 

1 

45.992 

base 

A\ 

+ 

10 

22. 484 

<y 

30 

12 

35.500 

Petit 

X' 

88 

25 

13.518 









-1 


§(<£+<£') 

o t n 

30 13 28 

s 

COS a 

4.229508 
9.282521 

s 2 

sin 2 a 

8.4590 
9.9837 

h 2 

4.047 

-h 

s 2 sin 2 a 

2.024 

8.443 


ft 

B 

8.511557 

C 

1.1711 

D 

2.331 

E 

5.917 

1st term 
2d,3d, and \ 
4th terms/ 
— A<f> 

+ 105.5810 

+ 0.4109 

+105.9919 

h 

2.023586 


9.6138 

+0.4109 


6.378 

+0.0002 


6.384 

-0.0002 


s 

sin a 

A' 

sec (ft’ 

4.229508 
9.991874 
8.509354 
0.063392 

A\ ’ 

sin J($+<£') 

2.794128 
9.701905 


2. 794128 


2. 496033 


ft 


ft 

A\ 

+ 622.4837 

—Ja 

+313.35 


STATION HORN 







0 

/ 

ft 

CL 

Petit to Grand 



177 

10 

20.4 

Third angle 

Grand and Horn 


- 80 

38 

49.8 

CL 

Petit to Horn 



96 

31 

30.6 

Aa 





— 

5 

29.2 






180 

00 

00.00 

a' 

Horn to Petit 



276 

26 

01.4 


O 

/ 

ft 






<t> 

30 

12 

35.500 

Petit 

X 

88 

25 

13.517 

A<f> 

+ 

1 

04.511 


A\ 

+ 

10 

54.021 

<t>' 

30 

13 

40.011 

Horn 

X' 

88 

36 

07.538 









-1 



O f ft 

30 13 08 

ft 

s 

cos a 

B 

4.245589 
9.055530 
8.511559 

S 2 

sin 2 a 

C 

8.4912 
9.9943 
1.1706 

h 2 

D 

3.625 
2.331 

-h 

s 2 sin 2 a 
E 

1.813 

8. 485 

5.916 

1st term 

2d,3d, andl 
4th terms / 
—A<{> 

-64.9648 

+ 0.4533 

h 

1.812678 


9.6561 
+0.4530 


5.956 
+0.0001 


6.214 
+0.0002 

-64.5115 










s 

sin a 

A' 

sec <j>' 

4.245589 
9.997178 
8.509354 
0.063471 

AX 

sin i(<£+<£') 

2.815592 
9. 701830 


2.815.592 


2. 517422 


ft 


ft 

A\ 

+654.0214 

—Act 

+329.17 





































































150 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Preliminary position computation , 

STATION PASCAGOULA 


Second angle 

a 

Joe 


4> 

j<f> 


1st term 
2d,3d,and \ 
4th terms/ 

— A<j> 


Grand to Horn 
Horn and Pascagoula 

Grand to Pascagoula 


Pascagoula to Grand 


30 


+ 


19 

1 


30 


30 20 00 

tt 

-81. 2312 
+ 0.2256 


20 


19. 589 
21.006 


40. 595 


First angle of triangle 
Grand 
Pascagoula 


X 

A\ 


o 

58 
+ 43 

/ 

14 

14 

n 

46.3 

20.1 

101 

29 

06.4 

— 

3 

52.8 

180 

00 

00.00 

281 

25 

13.6 

97 

55 

57.5 

88 

25 

36.522 

+ 

7 

40.922 


88 


33 


17. 444 


-81.0056 
s 

sin a 
A' 

sec <f>' 


s 

4. 099072 

«2 

8.1981 



-h 

1.910 

COS a 

9. 299100 

sin 2 a 

9.9824 

h2 

3.820 

s 2 sin 2 a 

8.180 

B 

8.511551 

C 

1.1725 

D 

2. 332 

E 

5.918 

h 

1.909723 


9. 3530 


6.152 


6.008 




+0.2254 


+0.0001 


+0.0001 


A\ 


4.099072 
9.991216 
8. 509351 
0.063988 

A\ 

sin K<f>+<f>') 

2.663627 


tt 


+460. 9215 

—Aa 


2.663627 
9. 703317 


2.366944 


+232. 78 


STATION BELLE 















o 

/ 

tt 

CL 

Pascagoula to Horn 







19 

21 

11.1 

Second angle 

Horn and Belle 









+ 69 

43 

24.6 

a 

Pascagoula to Belle 








89 

04 

35.7 

Act 













— 

5 

01.8 














180 

00 

00.00 

a' 

Belle to Pascagoula 








268 

59 

33.9 










First angle of triangle 

48 

58 

48.0 


o 



/ 



tt 









4> 

30 



20 

40.595 


Pascagoula 

X 

88 

33 

17. 444 

A<j> 

— 





8.730 





JX 

+ 

9 

57.323 

V 

30 



20 

31.865 
—1 


Belle 


X' 

88 

43 

14.767 


O / 

It 


s 


4.202937 

s 2 

8. 4059 




K*+tf') 

30 20 

36 

COS a 


8. 207256 

sin 2 a 

9.9999 








B 


8. 511550 


"t 

1.1729 




1st term 

tt 

+8.3511 

t 

L 


0.921743 



9. 5787 




2d term 

+0. 3790 








-j-u. o<yu 




— A<$> 

+8. 7301 














s 



4. 202937 











sin a 



9.999944 











A' 



8. 509351 


A\ 


2. 776209 






sec <J>' 



0.063977 

sin 

9. 703447 









2. 776209 




2. 479656 










1 






tt 






A\ 


+597.3226 


—Aa 


+301. 76 















































































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


151 


secondary triangulation —Continued 

STATION PASCAGOULA 







O 

/ 

// 

a 

Horn to Grand 



238 

09 

28.1 

Third angle 

Pascagoula and Grand 


- 38 

49 

42.8 

a 

Horn to Pascagoula 


199 

19 

45.3 

Jet 





+ 

1 

25.8 






• 180 

00 

00.00 

a' 

Pascagoula to Horn 


19 

21 

11.1 


O 

/ 

n 






<t> 

30 

13 

40. 011 

Horn 

X 

88 

36 

07.539 

j<f> 

+ 

7 

00. 584 


JX 

— 

2 

50.094 

4' 

30 

20 

40.595 

Pascagoula 

X' 

88 

33 

17.445 









-1 



o tn 

S 

4.137527 

S 2 

8.2750 



-h 

2.624 

30 17 10 

COS a 

9.974803 

sin 2 a 

9.0396 

h 2 

5.248 

s 2 sin 2 a 

7. 315 


n 

B 

8.511558 

C 

1.1709 

D 

2.331 

E 

5. 917 

1st term 

-420.6182 

h 

2.623888 


8. 4855 


7. 579 


5.856 

2d,3d,andl 
4th terms / 

+ 0.0345 




+0.0306 


+0.0038 


+0.0001 

-J4 

-420.5837 










s 

sin a 

A' 

sec <f>' 

4.137527 
9.519824 
8.509351 
0.063988 

J\ 

sin £(</>+<£') 

2. 230690 
9. 702706 


2.230690 


1.933396 


it 


rr 

JX 

-170.0944 

—Jot 

> -85.78 


STATION BELLE 












O 

/ 

if 

a 

Horn to Pascagoula 






199 

19 

45.3 

Third angle 

Belle and Pascagoula 





- 61 

17 

47.9 

a 

Horn to Belle 







138 

01 

57.4 

Jot 










— 

3 

35.5 











180 

00 

00.00 

a' 

Belle to Horn 







317 

58 

21.9 


O 


/ 


n 








4 

30 


13 

40. 011 


Horn 

X 

88 

36 

07. 539 

J 4 

+ 


6 

51.853 




J\ 

+ 

7 

07. 228 

4' 

30 


20 

31.864 


Belle 

X' 

88 

43 

14. 767 


O t 

it 

s 

4. 232096 

s 2 

8.4642 



-h 

2.615 

\{4^~4') 

30 17 

06 

COS a 

9.871296 

sin 2 a 

9.6505 

h 2 

5.230 

s 2 sin 2 a 

8.115 


it 


B 

8. 511558 

C 

1.1709 

D 

2.331 

E 

5. 917 

1st term 

-412.0501 

I 

L 

2. 614950 


9.2856 


7.561 


6.647 

2d,3d,andl 
4th terms / 

+ 0.1976 






+0.1936 


+0.0036 


+0.0004 

— J4 

-411.8525 












s 

sin a 
A' 

sec <f>' 


JX 


4.232096 
9.825236 
8.509351 
0.063977 

J\ 

sin U4+4') 

2. 630660 


it 


+427.2283 

—Jet 


2.630660 
9.702690 


2.333350 


+215.45 







































































152 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Preliminary position computation , 


STATION CLUB 








o 

/ 

// 

a 

Belle to Horn 




317 

58 

21.9 

Second angle 

Horn and Club 




+ 40 

30 

06.7 

a 

Belle to Club 




358 

28 

28.6 

Aa 






+ 


05.0 







180 

00 

00.00 

a' 

Club to 

Belle 




178 

28 

33.6 

0 




First angle of triangle 

105 

43 

56.5 


O 

/ 

tt 







30 

20 

31. 864 

Belle 

X 

88 

43 

14. 767 

A<f> 

— 

5 

19. 886 


A\ 

— 


09. 812 

<t>' 

30 

15 

11.978 

Club 

X' 

88 

43 

04.955 









+ 1 



O t tt 

5 

3. 993596 

S 2 

7.9872 




30 17 52 

COS a 

9.999846 

sin 2 a 

6.8504 

h 2 

5.010 

tt 

B 

8.511550 

C 

1.1729 

D 

2.332 

1st term 

+319.8836 

h 

2. 504992 


6.0105 


7.342 

2d and 3d\ 
terms / 

+ 0.0023 

# 



+0.0001 


+0.0022 

— 

+319.8859 








s 

sin a. 

A' 

sec <£' 

3.993596 
8. 425207 
8.509353 
0.063584 

JX 

sin 

0.991740 
9. 702856 


0.991740 


0. 694596 


ft 


ft 

JX 

-9.8116 

—Aa 

-4.95 


STATION DEER 


O 

t 

// 

358 

28 

28.6 

+102 

35 

20.0 

101 

03 

48.6 

— 

2 

45.3 

180 

00 

00.00 

281 

01 

03.3 



+ . 1 

41 

02 

11.4 

88 

43 

14. 767 

+ 

5 

27.103 

88 

48 

41.870 


Second angle 

a 
A a 


<i> 

Acf> 


Belle to Club 
Club and Deer 

Belle to Deer 


Deer to Belle 


30 


+ 


30 


20 


21 


31. 864 
55. 356 


27. 220 


First angle of triangle 
Belle 


Deer 


X 

A\ 

X' 


s 

sin a 
A' 

sec <p 


A\ 


3. 949436 
9.991853 
8. .509351 
0. 064045 


2. 514685 


+327.1033 


A\ 

sin £(<$+<£') 


—Jet 


2.514685 
9. 703531 


2.218216 
tt 

+ 165. 28 



o r tt 

s 

3.949436 

s 2 

7.8989 



*(*+*') 

30 21 00 

COS a 

9.283068 

sin 2 a 

9.9837 

h 2 

3. 488 

tt 

B 

8.511550 

C 

1.1729 

D 

2.332 

1st term 

-55. 4695 

h 

1. 744054 


9.0555 


5.820 

2d and 3dl 
terms / 

+ 0.1137 




+0.1136 


+0.0001 

— A<}> 

-55.3558 






































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 153 


secondary truing illation —Contin lied 

STATION CLUB 















O 

/ 

tt 

a 

Horn to Belle 









138 

01 

57.4 

Third angle 

Club and Belle 









- 33 

45 

57.1 

a 

Horn to Club 









104 

16 

00.3 

Jce 













— 

3 

30.2 














180 

(X) 

00. 00 

a' 

Club to Horn 









284 

12 

30.1 


O 



t 


tt 









<t> 

30 



13 

40.011 


Horn 

X 

88 

36 

07.539 

J<{> / 

+ 



1 

31. 967 





JX 

+ 

6 

57. 418 

<y 

30 



15 

11.978 


Club 

X' 

88 

43 

04.957 
















-1 


O / 

t 


s 

4. 061239 

S 2 


8.1225 



— h 

1.964 


30 14 

26 

COS a 

9. 391705 

sin 2 a. 

9.9728 

h 2 

3.929 

s 2 sin 2 a 

8.095 

tt 



B 

8.511558 

C 


1.1709 

I) 

2.331 

E 

5. 917 

1st term 

-92.1514 

h 

1.964502 



9.2662 


6.260 


5.976 

2d, 3d, and \ 
4th terms j 

+ 0.1848 








+0.1845 


+0.0002 


+0. 0001 

—J<}> 

—91.9666 














s 



4.061239 











sin a 



9.986395 











A' 



8.509353 


JX 


2.620571 






sec <j>' 



0.063584 

sin a(0+<£') 

9. 702113 









2.620571 




2.322684 









tt 






tt m 






JX 


+417.4179 


—Joe 

+210.22 






STATION DEER 


. 





o 

/ 

tt 

a. 

Club to Bello 



178 

28 

33.6 

Third angle 

Deer and Belle 



- 36 

22 

28.8 

a 

Club to Deer 



142 

06 

04.8 

Ja 





— 

2 

50.0 






180 

00 

00.00 

a ' 

Deer to Club 



322 

03 

14.8 


O 

r 

tt 






<t> 

30 

15 

11.978 

Club 

X 

88 

43 

04.956 

j<i> 

+ 

6 

15. 242 


JX 

+ 

5 

36.915 

<y 

30 

21 

27.220 

Deer 

X' 

88 

48 

41.871 


a(0+<£') 

O t tt 

30 18 20 

tt 

s 

COS a 

B 

4.165767 
9. 897131 
8.511556 

s 2 

sin 2 a 

C 

8.3315 
9.5767 
1.1713 

h 2 

D 

5.149 

2.331 

-h 

s 2 sin 2 a 
E 

2.574 

7.908 

5.918 

1st term 

-375.3652 

h 

2. 574454 


9.0795 


7. 480 


6. 400 

2d,3d,andl 
4th terms / 

+ 0.1233 




+0.1200 


+0.0030 


+0.0003 

— J<j> 

-375.2419 










S 

Sin a 

A' 

sec </>’ 

4.165767 
9. 788357 
8. 509351 
0. 064045 

JX 

sin £(<£+<£') 

2. 527520 
9. 702957 


2. 527520 


2. 230477 


tt 


tt 

JX 

+336.9f47 

—Ja 

+ 170.01 






































































154 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Preliminary 'position computation , 


STATION SHIP 


% 


0 

r 

// 

322 

03 

14.8 

+ 56 

47 

24. 7 

18 

50 

39.5 

— 

1 

14.8 

180 

00 

00. 00 

198 

49 

24.7 

70 

52 

34.5 

88 

48 

41. 870 

+ 

2 

28. 269 

88 

51 

10.139 


Second angle 

a 

Jot 


4 > 

j(j> 


4? 




1st term 
2d and 3d 
terms 

— J<f> 


Deer to Club 
Club and Ship 

Deer to Ship 


Ship to Deer 


30 


21 

6 


27.220 
17.200 


First angle of triangle 

Deer X 

J\ 


S 

sin a 

A' 

sec 

4.088917 
9.509200 
8. 509353 
0.063581 

J\ 

sin *(<£+<£') 


2.171051 



rt 


JX 

+148.2692 

—Jot 


2.171051 
9. 702952 


1. 874003 


+ 74.82 


30 

15 10.020 

Ship 


X' 

O /ft 

5 

4.088917 

S 2 

8.1778 


30 18 19 

COS a 

9. 976075 

sin 2 a. 

9. 0184 

h 2 

n 

B 

8.511549 

C 

1.1731 

D 

+377.1734 

h 

2. 576541 


8. 3693 


+ 0.0265 




+0.0234 


+377.1999 







5.153 
2. 332 


7. 485 
+0. 0031 


STATION BILOXI LIGHTHOUSE 








0 

t 

tt 

a 

Deer to Ship 




18 

50 

39.5 

Second angle 

Ship and Biloxi Lighthouse 


+ 96 

30 

33.5 

a 

Deer to Biloxi Lighthouse 


115 

21 

13.0 

Jot 






— 

2 

42.9 







180 

00 

00. 00 

a ' 

Biloxi Lighthouse to Deer 


295 

18 

30.1 





First angle of triangle 

48 

11 

21.0 


O 

30 

/ 

21 

tt 

27. 220 

Deer 

X 

88 

48 

41. 870 

j<i> 

+ 

2 

12. 200 


J\ 

+ 

5 

22. 085 

P 

30 

23 

39. 420 

Biloxi Lighthouse 

X' 

88 

54 

03. 955 




-1 





-1 



o t tt 

30 22 33 

$ 

COS a 

3.978395 
9. 631650 

s 2 

sin 2 a 

7.9568 
9.9120 

h 2 

4. 243 

-h 

s 2 sin 2 a 

2.121 
7.869 

tt 

B 

8. 511549 

C 

1.1731 

D 

2.332 

E 

5. 919 

1st term 

2d, 3d, and 1 
4th terms / 

— J<j> 

-132.3104 

+ 0.1107 

-132.1997 

h 

2.121594 


9.0419 

+0.1102 


6.575 

+0. 0004 


5.909 

+0. 0001 


s 

sin a 

A' 

sec tj>’ 

3. 978395 
9.956016 
8. 509350 
0.064209 

J\ 

sin U<t>+</>') 

2. 507970 

9. 703868 



2. 507970 


2. 211838 



tt 


n 


J\ 

+322.0846 

—Jot 

+ 162.87 

• 









































































APPLICATION OP LEAST SQUARES TO TRIANGULATION. 155 


secondary triangulation —Continued 


STATION SHIP 








o 

/ 

tt 

CL 

Club to Deer 




142 

06 

04.8 

Third angle 

Ship and Deer 




- 52 

20 

01.2 

a 

Club to Ship 




89 

46 

03.6 

Ja 






— 

4 

04.4 







180 

00 

00.00 

a' 

Ship to Club 




269 

41 

59.2 


O 

t 

ft 






4> 

30 

15 

11.978 

Club 

X 

88 

43 

04. 956 

J<f> 

— 


01.958 


JX 

+ 

8 

05.182 

<t>' 

30 

15 

10.020 

Ship 

X' 

88 

51 

10.138 








+ 1 


£(<£+<£') 

o t it 

30 15 11 

ft 

s 

COS a 

B 

4.112975 
7.607988 
8.511556 

s 2 

sin 2 a 

C 

8.2260 
0.0000 
1.1713 

1st term 

+ 1.7081 

h 

0. 232519 


9. 3973 

2d term 

+0. 2496 




+0. 2496 

-A4> 

+ 1.9577 






s 

sin a 

A' 

sec <f> ' 

4. 112975 
9.999996 
8.509353 
0.063581 

JX 

sin i(^+0 > ) 

.2.685905 
9. 702275 


2.685905 


2. 388180 


It 


ft 

JX 

+485.1824 

—Ja 

+244. 44 


STATION BILOXI LIGHTHOUSE 








o 

t 

ft 

CL 

Ship to Deer 




198 

49 

24.7 

Third angle 

Biloxi Lighthouse and Deer 


- 35 

18 

05. 8, 

CL 

Ship to Biloxi Lighthouse 



163 

31 

18.9 

Ja 






— 

1 

27.8 







180 

00 

00. (K) 

a ’ 

Biloxi Lighthouse to Ship 



343 

29 

51.1 


O 

t 

ft 







30 

15 

10.020 

Ship 

X 

88 

51 

10.139 

Jcf> 

+ 

8 

29. 399 


JX 

+ 

2 

53. 815 


30 

23 

39. 419 

Biloxi Lighthouse 

X' 

88 

54 

03.954 


Fixed value 

39.419 





06. 820 



o r tt 

s 

4.213748 

s 2 

8. 4275 



KW) 

30 19 25 

COS a 

9.981786 

sin 2 a 

8.9056 

h 2 

5.414 

tt 

B 

8.511556 

C 

1.1713 

D 

2.331 

1st term 

-509.4364 

h 

2. 707090 


8.5044 


7. 745 

2d,3d,andl 
4th terms / 

+ 0.0376 




+0.0319 


+0.0056 

-J<j> 

-509.3988 








-h 

«2 sin 2 a 
E 


2. 707 
7.333 
5. 918 


5.958 

+0.0001 


s 

sin a 

A' 

sec <£' 

4. 213748 
9. 452781 
8.509350 
0. 064209 

JX 

sin £(<£+<£') 

2.240088 
9. 703190 


2. 240088 


1.943278 


tt 


tt 

JX 

+ 173. 8153 

—Ja 

+87. 76 






































































156 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

Preliminary position computation , 


STATION SHIP ISLAND LIGHTHOUSE 








0 

t 

tt 

a 

Biloxi Lighthouse to Ship 


343 

29 

51*. 1 

Second angle 

Ship and Ship Island Lighthouse 


+ 33 

44 

13.3 

a 

Biloxi Lighthouse to Ship Island Lighthouse 


17 

14 

04.4 

Ja 






— 

1 

57.9 

• 






ISO 

00 

00. 00 

a' 

Ship Island Lighthouse to Biloxi Lighthouse 


197 

12 

06.5 





First angle of triangle 

50 

31 

40.3 


O 

30 

t 

23 

" 

39. 419 

Biloxi Lighthouse 

X 

88 

54 

03. 954 

Ja 

— 

10 

54.077 


JX 

+ 

3 

53. 590 

<t>' 

30 

12 

45.342 

Ship Island Light- 

X' 

88 

57 

57.544 




+1 

house 







O t tt 

30 18 12 

tt 

s 

COS a 

B 

4.323989 
9.980049 
8.511546 

«2 

sin 2 a 

C 

8.6480 
8.9434 
1.1738 

h 2 

D 

5.631 

2.332 

-h 

s 2 sin 2 a 
E 

2.816 

7.592 

5.919 

1st term 

2d,3d, andl 
4th terms j 
— J<f) 

+654.0094 
+ 0.0673 

h 

2. 815584 


8. 7652 

+0.0583 


7.963 

+0.0092 


6.327 

-0.0002 

+654.0767 










s 

Sill a 

A' 

sec <£' 

4.323989 
9. 471708 
8. 509354 
0.063404 

JX 

sin J 

2.368455 
9. 702930 


2.368455 


2.071385 


tt 


tt 

JX 

+233. 5904 

—Ja 

+117.87 


Fixed a Biloxi Lighthouse to Ship Island Lighthouse, 17° 14 7 17.6". 



































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 157 


secondary triangulation —Continued 

STATION SHIP ISLAND LIGHTHOUSE 








O 

r 

ft 

tt 

Ship to Biloxi Lighthouse 



163 

31 

18.9 

Third angle 

Ship Island Lighthouse and Biloxi Lighthouse 


— 95 

44 

06.9 

CL 

Ship to Ship Island Lighthouse 


67 

47 

12.0 

Joe 






— 

3 

25.1 







ISO 

00 

00. 00 

a' 

Ship Island Lighthouse to Ship 


247 

43 

46.9 

-.1 


O 

30 

/ 

15 

ft 

10.020 

Ship 

X 

. 88 

51 

10.139 

J<j> 

— 

2 

24. 677 

JX 

+ 

6 

47. 405 

V 

30 

12 

45.343 

Ship Island Light- 

X' 

88 

57 

57.544 




house 






Fixed value 

45. 341 





57. 464 


1st term 
2d, 3d, and! 
4th term / 

— J<f> 


30 13 58 

tt 

+ 144. 5010 
+ 0.1764 


+ 144.6774 


s 

4.070759 

s 2 

8.1415 


COS at 

9. 577556 

sill 2 a 

9.9330 

h 2 

B 

8.511556 

C 

1.1713 

D 

h 

2.159871 


9. 2458 

+0.1761 



4.320 

2.331 


6.651 
+0.0004 


-h 

s 2 sin 2 a 

E 


2.160 
8. 075 
5.918 


6.153 
- 0.0001 


S 

sin a 

A' 

sec (J> r 

4.070759 
9.966509 
8.509354 
0.063404 

JX 

sin £($+<£') 

2.610026 
9. 702011 


2. 610026 


2.312037 


ft 


rr 

JX 

+ 407. 4046 

—Jot 

+205.13 



Fig. 7. 

















































158 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

FORMATION OF OBSERVATION EQUATIONS 

The position computation was carried westward from the fixed 
lines at the eastern end of the scheme and the observation equations 
were formed in the same order. The treatment of fixed lines is the 
same as in the first adjustment of figure 6, pages 105 et seq. No 
new detail arises until the points Deer and Ship are reached, which 
have lines connecting them with the fixed points Biloxi Lighthouse 
and Ship Island Lighthouse. Suppose for the moment that Biloxi 
Lighthouse were not fixed but that its latitude and longitude were 
to receive corrections of $0 12 and <^ 12 respectively. The observation 
equation for v 57 would then read, 

v 57 = z 13 — 603$4* 248$I n 4- 603$0 12 — 248$1 13 4~ 0.5 

The latitude and longitude of Biloxi Lighthouse as developed by 
the preliminary position computation are 30° 23' 39."419 and 88° 
54' 03."954, while the fixed values are 30° 23' 39."419 and 88° 54' 
03."820, so that to reduce the preliminary to the fixed values on 
which the adjustment is built corrections of $<jS 12 = 0.000* and 
$1 12 = —0.134 are necessary. By substituting these values in the 
equation for v 57 there results, 

^57 = ^i3-603$^ 11 +248$l 11 + 603 (0.000)-248 (-0.134)4-0.5 
or, 

v 57 = Si 3 - 603$<£ n 4- 248$I n 4-33.7 
as given in the table page 16. 

The equation for the reverse direction, v 63 , if Biloxi Lighthouse 
were not fixed, would be, 

= s 15 - 603 d(f> n + 248$4 + 603$<£ 12 - 248$I 12 4-0.0 
which with the use of the above values of $<jf> 12 and $I 12 becomes, 
v 63 = 2is - 603$<£ n + 248$l n 4-33.2 

Similar computations must be made for v 56 , v 58 , v 59 , v 64 , v 87 , and v^. 

The known terms in the expressions for v Q5 , and may be found 
in a similar way by allowing for the fixity of both ends of the line. 
If the corrections needed to the preliminary position of Ship Island 
Lighthouse in order to reduce it to the fixed position are called $</> 13 
and $I 13 then for the line Biloxi Lighthouse to Ship Island Lighthouse 
the formula gives, 

da = 89 ($ cf> 13 — $0 12 ) 4- 250 ($I 13 -$4) 

If the line were free to be turned in azimuth, then by the adjustment, 
= 2 15 + 89($<£ 13 - $& 2 ) 4- 250($I 13 - $4) -1.7 


* This zero discrepancy is merely accidental. 




APPLICATION OF LEAST SQUARES TO TRIANGULATION. 159 


But to reduce the positions of the preliminary computation to 
the fixed positions, d(f> 13 = -0.002, dcj> l2 = 0.000, > 4 ,=-0.080, and 
^ 12 = —0.134 (see pp. 155, 157). Substituting these values gives, 


/y 65 = 2 i5 + 11-6 

as given in the table. is found by a similar process. 

The effect of using for the preliminary computation values from a 
previous adjustment for the figure but not for the positions appears 
in the constant term of the normal equations until the effect of 
closure in positions comes in. These constant terms should be zero 
except for the effect of accumulated errors in the last place of the 
two computations and a slight difference in the treatment of direc¬ 
tions (3) and (3a). They are in fact almost negligible until the effect 
of closure appears in the equation for dcf> s , from which point they 
become quite large. 


Fort Morgan-Dauphin Island east base: 

o / // 

Assumed azimuth, 101 45 44.9 
Observed azimuth, 101 45 44.d—z l +v l 

0 = 0 . 0 J rz l —v l 

i; 1 =2r 1 -}-0. 0 


Fort Morgan-Cedar: ° / " 

Assumed azimuth, 144 19 22. 9+dc* 

Observed azimuth, 144 19 22.6 —z l -\-v 2 

0=+0. 3+z 1 -r’ 2 +229<?0 I -277(W 1 
v 2 =z t +229<?0 1 -277o/> 1 +O. 3 


DAUTHIN ISLAND EAST BASE 


Assumed 

azimuth 

Observed azimuth 

Equation 

Station observed 

O / // 

84 14 41.9 
137 52 06.3 
188 10 17.4 
281 42 17.9 

O t ft 

84 14 41.9— Z 2 +V 3 

137 52 O 6 . 9 -z 2 +04 

188 10 18.3-z 2 +0 5 

281 42 17.9—Z 2 +V 30 

03 =Z2+0.0 

04 = +Z2+4205$2 — 4035X2 — 0.6 

05 =+Z 2 —825$i—4985Xi—0.9 

030 = 22 + 0.0 

West base 

Cat 

Cedar 

Fort Morgan 


(1) CEDAR 


324 16 25.0 

324 16 25.0—Z 3 + 04 O 

04o=23+229J<^i— 277d\j+0.0 

Fort Morgan 

8 10 46.8 

8 10 46. 8 -Z 3 + 05 O 

050 = Z 3 —82<J < pi —4985 Xi+0.0 

East base 

45 37 18.9 

45 37 19. 8 -Z 3+06 

06 =z 3 —266 (?<£i— 226<?Xi—0.9 

West base 

68 22 02 .4 

68 22 OI.O-Z 3+07 

Vi — Z 3 — 657d<£i — 226dXi+657<?<jS2 - t - 22t)d Ai»+1.4 

Cat 

99 17 00.4 

99 17 02. 5-Z3+08 

08 =23 —341<J0i+ 485 Xi+34 ld 03 — 485X3—2.1 

Pins 


( 2 ) CAT 


23 51 22.0 

23 51 22.0-Z4+017 

017= Z4—2735(^2—5375X2+0.0 

West base 

122 37 00. 7 

/ 

Pins 

248 19 24. 4 

248 19 27.1-Z4+01O 

V \ s = Z \ —6575<^i—2265 Xi+6575^2+2205X2—2.7 

Cedar 

317 49 57.9 

317 49 59.8-Z4+016 

016= 24+ 4205^2—4035X2—1.9 

East base 


DAUPHIN ISLAND WEST BASE 


264 11 22.1 

264 11.22.1—Z5+014 

014= 25+0.0 

East base 

78 57 02.1 

78 57 01. 7—25+09 

09 =25+3675(^4+625X4+0.4 

Petit 

118 03 41.5 

118 03 4O.8-Z5+01O 

010=25+2875^5—1335X5+0.7 

Grand 

157 58 33.5 

157 58 31. 7-Z5+011 

011=25+1485(^3—3I85X3+I.8 

Pins 

203 50 10. 3 

203 50 IO.9-Z5+012 

012=25—2735^2—5375X2—0.6 

Cat 

225 33 29. 4 

225 33 29.6-Z5+013 

013= Z5—2665 4> i —2265 X i—0.2 

Cedar 


91865°—15-11 

























160 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


(3) PINS 


Assumed 

azimuth 

Observed azimuth 

Equation 

Station observed 

O f tt 

279 11 16. 6 
302 33 55.1 
337 56 39. 8 
30 22 49.2 
63 09 48.1 

o r ft 

279 11 16. 6-z 6 +#i 8 
302 33 56.9—2e+#i9 
337 56 42.6—Ze+#2o 

30 22 52.1 — 26+#2i 

63 09 49.6 — 2e+#22 

#is=Z6 — 341501+ 485Xi+341503— 485X3+0.0 
# 19 = Ze—459502+2555X2+459503 —2555 X 3 —1.8 
#20=26+148503 —3 I 85 X 3 —2.8 
# 21 = Z6 — 1535 <$3— 2285X3+1535</>4+2285X4—2.9 
$22=26 — 450503 — I 985 X 3 +450505+1985X.5 — 1.5 

Cedar 

Cat 

West base 

Petit 

Grand 

(4) PETIT 

96 31 30. 6 
139 08 41. 2 
177 10 20. 4 
210 19 28.8 
258 51 48. 7 

96 31 30. 6—Z 7 +V 28 
139 08 40.9—Z 7 +W 29 

177 10 19.5—Z 7 +V 30 
210 19 28.2-z 7 +i>3i 
258 51 47. 7 -Z 7 +V 32 

$ 28 = 27 —358504+365X4+358506—365X6+0.0 
$ 29 = 27 — 209504 + 2115 X 4 + 209507 — 2115 X 7 + 0.3 
$ 30 = z^ — 25504 +4425X 4+25505— 4425X 5 +0.9 
$3i=Z7—153503 — 2285X 3+ 153504 + 2285X 4+ 0.6 
$32=27+367504+625X4+1.0 

Horn 

Pascagoula 

Grand 

Pins 

West base 

(5) GRAND 

243 06 15.8 
297 58 16.0 
357 10 08. 8 
58 14 46.3 
101 29 06.4 

243 06 15.8-Z8+023 
297 58 17. 4-z 8 +# 2 4 
357 10 09. 4 -z 8 +i>25 

58 14 46.6—Zs+#26 
101 29 05.5-z 8 +#27 

$23=28 — 450503 — 1985X3 +450505+1985X5+0.0 
$ 24 = Z 8+287505— 1335X 5 — 1 .4 
# 25 = 28— 255 04 + 4425X 4 +2550s — 4425X 5— 0.6 
#26= 28—272505— 1465X 5 +27250<; + 1465X6 — 0.3 
$ 27 = Zs — 4955 05 + 875 X 5 +495507— 87 5 X 7 +0.9 

Pins 

West base 

Petit 

Horn 

Pascagoula 

(6) HORN 

104 16 00. 3 
138 01 57. 4 
199 19 45.3 
238 09 28.1 
276 26 01.4 

104 16 00. 3 -Z 9 +V 38 
138 01 54.0-z 9 +# 3 9 
199 19 47.5—z 9 +t>40 
238 09 26. 5 —Z 9 +V 41 
276 26 00. 7 -Z 9 +D 42 

#38= 29— 535506+1185X6+535508 — 1185X8+0.0 
# 39=29—249506 + 2405 X 6+249509— 2405X9+3.4 
$ 40 = Zg+153506+3795X6 —153507 — 3795 X 7 —2.2 
$ 41 = Z 9—272505— 1465 X 5 + 272506 + 1465 X 6 + 1.6 
$ 42 = 29— 3585 04 + 365 X 4 +358506 — 365X6+0.7 

Club 

Belle 

Pascagoula 

Grand 

Petit 

(7) PASCAGOULA 

281 25 13.6 
319 04 37. 2 
19 21 11.1 
57 14 14.4 
89 04 35. 7 

281 25 13.6 — Z 10 +V 33 
319 04 34. 2 - 210+^34 
19 21 07. 6 — Zio +^35 
57 14 12.2—Zio+t>36 
89 04 35.6 — Z 10 +V 37 

$ 33 = 210 — 495505 + 875 X 5+495507 — 875X7+0.0 
#34=210— 209504+2115X4+209507— 211dX7+3. 0 
#35=2io+153506+3795X6 —153507—3795X7+3. 5 
#36= 210 - 286507-1605 X 7 +286508+1605X 8 + 2.2 
# 37 = 210 —398507-65X7+398509+65X 9 +O.l 

Grand 

Petit 

Horn 

Club 

Belle 

(9) BELLE 

268 59 33.9 
317 58 21.9 
358 28 28.6 
52 04 22. 8 
101 03 48.6 

268 59 33.9— zn+#43 
317 58 23. 7— Zn +#44 
358 28 29.2-Zn+#4 5 
52 04 24. 8 —Z 11 +W 46 
101 03 47.7— Zn+i;47 

#43=211 — 398507 — 65 X 7 +3985 09 + 65 X 9 + 0.0 
# 44 = Z11 —249506+ 2405X 8 +249509 — 2405X 9 —1.8 
#45= 2 n— 17508 + 5595 X 8 + 17509 — 5595 X 9 —0.6 
# 46 = 2 n —311509— 2 H 5 X 9 + 3 II 501 O+ 2 II 5 X 1 O—2.0 

# 47=211 — 700509 + 1195 X 9+7005011 — 1195Xn+0.9 

Pascagoula 

Horn 

Club 

Ship 

Deer 

(8) CLUB 


89 46 03.6 

89 46 03.6—Zi2+#4 8 

#48=212 —490508—25X8+4905010+ 25 X 40 +0.0 

Ship 

142 06 04. 8 

142 06 07.1—zi2+#49 

# 49 = 212 —266508+ 2975X8+ 26650H—2975Xn—2.3 

Deer 

178 28 33.6 

178 28 33.9—£ 12+^50 

^50=212-17^8+559^ Xg+17$09—559$ X 9 —0.3 

Belle 

237 09 18.0 

237 09 17.2—Zi2+#5i 

# 51 = 212 -286507-1605X7+286508+1605X8+0.8 

Pascagoula 

284 12 30.1 

284 12 30. 8 — Zi2+#52 

#52=2i2-535506+1185X 6 +535508-1185X8-O.7 

Horn 



































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 


161 


(11) DEER 


Assumed 

azimuth 

Observed azimuth 

Equation 

Station observed 

o f ft 

281 01 03.4 
322 03 14. 8 
18 50 39. 5 
42 46 50. 3 

115 21 13.0 

Off/ 

281 01 03.4-2i 3 +t; M 
322 03 14.1-2 13 +t>5< 
18 50 41.3— 213+D55 
42 46 46. 1—Zi3+V56 

115 21 12.5—213+^57 

t?53=2i3— 700(J<^9+119^X9+700(J<^ii —119#Xn+0.0 
V;>i— Zi3 — 266cJ<j&8+297dX8+266(?^n — 297# Xu+0.7 
Vbb— Zi3+167d0io+425dXio — 167#^n—425# Xu—1.8 
^ 56 = 213 —197#0n—185#Xu—11.0 

Vbi = Z13 —603#$n+248#Xn+33.7 

Bello 

Club 

Ship 

Ship Island 
Lighthouse. 
Biloxi Lighthouse 


(10) SHIP 


67 47 12.0 

163 31 18.9 
198 49 24. 7 
232 00 23.1 
269 41 59. 2 

67 47 12.0—2H+V58 

163 31 19. 0—214+^59 
198 49 23.4 -2U+U60 
232 00 23. 6-214+Voi 
269 41 58.4—2i4+r 6 2 

Vbs— 2i4—500#(£io—177#Xio—15.2 

Vb9= 214-110#<jho+323#X 10 +43.2 

Vtio — 214 +167#0io+425# Xio— 167#^n — 425#Xn+1.3 
1>61=Z14—31U<£9—211#X 9 +311#<£io+211#X 1 o— 0.5 
Vf>2— 2i4—490#<^8 — 2 # X8+49O#0io+2#Xio+O.8 

Ship Island 
Lighthouse 
Biloxi Lighthouse 
Deer 

Belle 

Club 

BILOXI LIGHTHOUSE 

295 18 30.1 
343 29 51.1 
17 14 04. 4 

295 18 30. l-sis+res 
343 29 47. 5-2i 5 +ro4 
17 14 06. 1—2i5+l>65 

^ 63 = 215 —603#(£n+248#Xn+33.2 

Vsi= Zib — 110#<£io+323# Xio+46.9 

065— 215+11.6 

Deer 

Ship 

Ship Island 
Lighthouse 

SHIP ISLAND LIGHTHOUSE 

197 12 06. 5 
222 42 10.0 
247 43 46.8 

197 12 06.5—ZiG+tfea 
222 42 08.6-2 ig+?; 67 
247 43 47. 7—2i6+V68 

066= 216+13.3 

067= 2i6—197#<£n—185#Xn—13.8 

V68 = 2i6—5OO#0io—177#Xio—16.1 

Biloxi Lighthouse 
Deer 

Ship 


In order to get the quantities on a better relative basis, it is best 
to adopt lOOd^, 100£l u etc., as unknowns in the equations. The 
coefficients throughout will then be divided by 100, and from the 
solution we shall determine one hundred times the corrections in 
seconds to the various latitudes and longitudes 



















Table for formation of normals, 


1G2 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


f 
















































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


163 


•m 

r-H H »-H H r-H t •» 

T 

© 


O 'S' <D CO 05 'll 

o’h'o'oo'h 

1 1 I + 1 



N N 

00 00 

1 l 

+495 

+495 

+146 

+146 

r- n 
<N <M 

+ + 

00 CO O CO 

© CO ^ ^ 00 CO 

r-H r-H *-H ^rfl 

+ 1 1 1 + 1 

+450 
+287 
+ 25 
-272 
-495 
- 5 

+442 

+442 

»o iO 

<N CS 

I 1 

- 198 

- 198 

O O 

iO »o 

1 1 
































Table for formation of normals , No. 1 —Continued 


164 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


a 


rH tH tH rH tH f — rH tH r-H rH tH h'* 


rH rH tH tH iH I ■ 


t — H rH rH tH tH f ' 




O Tf* CM co lO 
O CO CM* H o CO 

+ I + + + 


00*0 CM rH 00 
© CO CO* CM* © od 

+++++ 


o oo«© o 05 *o 

© tH O* CM* O* CO 

I I I + I 


o co co oo *o 
o' wood ci 

I I + I I 


ONOOONO 


CM CM CO *0 CO CO 

■o' co h o o o 

HTJ1 CM 

+ I I ++ I++I++ 


05 05 

rH H 
tH iH 

I I 


u- C5 *o *o oo oo 

05 tH 05 CM 00 ^ 

CM tH CM Tjl tH CM tH 

I I I I I + I 


»o 


S O co 

o co 

cm 

++ + 


CO O O N N CO H 
CO O CO co 05 O 
CM NINHHO 

+ ++I I I I 


CM CM 
+ + 


+ 


'+ 


+ + 


IC CO ic 

<M r^(M <M ^ 00 

i co 'f in r-~ 




00 CO 
+ + 


+ 


+ 


+ 


t'- OONHOCO 
CO OHCOHOIO 
1“H iCHrlCO^CO 

+ I I++++ 


CO CO 

++ 


©OOHOJIO 
f lO H H DC 
NIONHOO 

+ 1 I I + I 


s 


I + 


05 

tH 

rH 

+ 


IM <N 

I I 


'o 


05 

s 

+ 


cm 

+ 


00 00 
05 05 

co co 

++ 


00 05NHON 
riHOl 1 

co cm CO CO 

+++1 I I 


+ 


+ I 


+ + 


WN05000CO 
g5 lO CO tH ~ 


+ + I+++I+ 


s a 


+ 


a 

+ 


I I 


* 0 - 

<J o 


lO 

CC 

*o 

+ 


+ 


+ + 


U- o CO N co lO 00 

rH OOHQOCO-^ 

CM Ol to 

I III+++ 


o o 

05 05 

T}1 ^ 


I I 


NH05< 
OOHNC 
CM CO t 


III I 


»C 05 CO CO 00 CO 
05 O »0 00 05 CO 
- < CM CO t ' 


I ++1 I I I I 


*©• 

'o 


00O05C0C0N 
H H N H CO ^ 
H(NCOrr 00 

++++1 + 


+ + 


o 

Si 

+ 


<N 

+ 


00(30 

tH »H 
rH rH 

++ 


kO 05 CO CM 00 tH 
CO H O N lO 
»C CM tH CM CO 

I 1+++1 


+ + 


KOKO 
00 CO 
ic 1C 


-©• 


oo 

+ 


oo 

+ 


++ 


<N 

+ 


<N 

+ 


00 00 
ic lO 
CO CO 


t rH *Q ® rH rH tH rH rH »0 ^ r 


1 ‘O «HHHHH»0 WH i 


I ‘O ^ rH rH rH t 











































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 165 
























Table for formation of normals 




166 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 



Oo'mrto'NOOONOONOOCoiNOi^OONN OQOOOO OOOat-. '^OOOtO 

HHT}iN»0' , TOOOO , l 1 HNHt'-l'»C^OC^Hl'»Hl>.0<X‘005 l ONOCOOCO<NHOH«C>CCOJO 

dddcocodtd>dH*diooocsHN^(N6ooi6(Nd r4 ^cNH^oddo^c^ddoo^o 

i i i i i i i i + i i i i i i +++ in i i i i i +++++ +1 i ++ 


MNO05NOO01'fHNONffiS^MXi©NXO000C05X5OO«0J®ONO'>t , CiC<30)b 

oddddddo'HiNddfiHijio'drtddddHNdH^ddooHrto'HOOoo 

+ + 1 1 1 1+11 III+ + +11+ 11111 +++++ 111+1 

r—1 

✓< 

-2.11 

-0.94z 

-0. 87 

—0.39i 

CO 

rH 


05 CO »2 

o 05 ®N 

CS O -r" CS 

+ + + + 

<N 

*© 

e< 

• co <S CO >S 

00 rH -S’ CO 

O O t-5 O 

1 1 + + 

H 

rH 

*o 

-e- 

+3.58 

+1.60* 

+2. 72 

+1.22* 

O 

T—( 

e< 

co *■?? oo o? cs oo co co cs co *S 

CO 1C 05 00 'S' 05 05 CO 'S' 'S' 00 05 

rH O HO ■*+’ 

1 1 ++ 1 1+111+1 

o> 

•e- 

+2.87 

+1.17* 

+4.50 
+2.01* 

+0.25 

+0.11* 
+4.50 
+2. 87 
+0.25 
-2. 72 
-4. 95 

-0.02* 

00 

/< 

CS IS 00 <N CO rH CS 00 CS 00 CS 00 

CO CS CS O CO rH 'S' CS CO CO 'S' 05 

O O CS rH O CS "S'CS ©'S’ + rH 

+ + + +++++++ + + 


‘0- 

© CO OOOOOJiCWNCS IO th 

CO 40 40C0 40 0CS400CO 04 rH 

CO rH rH o CO <N O rH CO o © © 

+ + + +111++1 1 1 

CO 

CO 

/< 

GC H 00 © oo 40 00 00 00 00 00 <N 00 <3 

TT C4 1 H co »C rH 04 © CD 04 005 00 

OO 00 H O C8 CO (N H H 04 HH © 

II 1 1 1 1 1 1 1 1 III 1 

40 

CO 

*0. 

rH C4 GO OHOSOOCOOH CO 00 Q h 

TT 40 Tfl CO ^ 40 ^ 40 40 40 40 <0*0 © 

CO rH H d CO ■H H H rf H rH © Tfl C4 

+ + + + + + + 11+ III 1 

Hfl 

Cl 

✓< 

-4.03 

-2.02* 

+2.26 

+1.01* 
-5. 37 
+2. 26 
-4.03 
-4.12* 

-5.37 

-2.19* 

+2.55 

+1.14* 

CO 

N 

'O. 

+4.20 

+2.10* 

+6.57 

+2.94* 
-2. 73 
+6. 57 
+4.20 
+ 4.64* 

-2.73 

-1.11* 

-4.59 

-2.05* 

<M 

< 

-2. 77 
-1.96* 

-4.98 
-2. 49* 
-2.77 
-4.98 
-2.26 
-2.26 
+0. 48 
—5.27* 

-2.26 

-1.30* 

-2.26 
-0.92* 
+0. 48 

+0.21* 

rH 


05 CS CSiHOSNCONiHOS 05 CO 05 rH Os 

CS CO OO'S'CSOOCO‘O'S'05 50 N COOS' o 

CS rH OOCSOCSCOCO'S' CO CO CS rH CO rH 

++ II + 1 1 1 1 1 1 1 III 1 


* All values in this table except those in the i) and I columns have been divided by 100. 














































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 167 



^2SSP 1 ;^,2 , ?? 0 "^S 0 2 0, ''_ c,000 'n OONioi-l'MOQOiO'OM go nnS 

O’l'N®NU5OO‘0INH0)O00®O0>'0OMM*NHOl'«)0CHMa«M300H©O®OC0tf)00® 

OMNHdrldriMoioMdHdNdHdfiddoHddHTiidffl'HiOHddHoioiHM'wNiNio' 

HM d -H* HN'fl'rtlOHHNH 

+1+++ + + + + + 11l+l ll + ll +11+ +1+ +1+++++++111 


0'fMON®00>ONH®000®OOJ>§OMMOONrtOt'OOON3wNM‘OOOc5lNO>®0)e<5«H(!o 
ocotNHdHdwwcidcodHowdHdddddHddHHco . 10 co h o d cococo h cs coco^ . 

HC0 05H^r HCO^H»OHHHC5 

+ 1+++ +++++ IIl+l ll+ll +11++1++1+++++++111 

CM 

£ 

-1.19 

-0.53 / 

-2.97 

-1.33/ 

-1.19 

-2. 97 

-4.25 

-1.85 

+2. 48 

-3. 48/ 

-4.25 

—1.90 / 

+2. 48 

+1.43/ 

-1.85 

-1.07i 

rH 

CM 

-& 

'o 

+7.00 
+3.13/ 

+2.66 

+ 1.19/ 
+7.00 

+2.66 

-1. 67 

-1.97 

-6.03 

-1.67 

—0. 75/ 

—6.03 

-3.48/ 

-1.97 

-1.14/ 

8 

o 

/< 

'O 

+2.11 

+0.94/ 
+0.02 

+0.01/ 

+4.25 

+1.90/ 

-1.77 

+3.23 

+4.25 

+ 2.11 

+0.02 

+3.50/ 

+3.23 

+1.86/ 

-1.77 

-1.02/ 

o> 

H 

5^io 

rH Q <55 r» OQOt^rHQOOCO Q 

rH CO rH CO N O H <0 H $ CO rH O COO 

CO rH Tf? <N rH 6 ^ H H CO "T H rH O 04 

+ ++ + + +11++++ II II 

oo ‘ 

rH 

o» 

/< 

*0 

-2.40 

-1.07/ 

+0.06 
+0.03/ 
+0.06 
-2. 40 
-5.59 
-2.11 
+ 1.19 
-3.96/ 

-5.59 

—2.50/ 

+ 1.19 

+0.53/ 

-2.11 

-0.94/ 

t>. 

H 

o» 

‘O* 

Oi rH OOOOXONHQO U- 00 O 0? H 5 

Tfl i-4 OlNOi^HHO»Q rH OO rH rH CO 

d i-t • coi-5cocl©coi^-.-i o on co co i-5 

+ + +++++III + +1 1 II 

o 

H 

00 

/< 

oo co o cs a q <n a o oo 35 co cs 7 h 

rH ot^- »Q iOOO^OcOrHOO CO oo 

rH o rH O *C CS O (N »0 H H rp cl rH OO 

1 1 + + + +1+++1+ + + II 

»o 

rH 

00 

'O. 

+5.35 

+2.39/ 

+2.86 

+1.28/ 

-0.17 

-0.08/ 
-4.90 
-2.66 
-0.17 
+2. 86 
+5. 35 
+0.21/ 

-2.66 

-1.19/ 

-4.90 
-2.19/ 

-V 

rH 

r» 

/< 

os aNHffiooi"® co o cl 

t- ©ooHNeot'O © co i-- 

co i-5ddco.-5©co© © i-5 o 

1 1 1 1 1 1 1 1 1 1 II 

CO 

tH 


co oo icacooco 05 oo oo co oo 

*(5 0050*00005*005 t— 00 d 

d n 5 d i-5 d co d co i-5 ci h 

1 1 ++ 1 1 1 1 1 1 II 

cs 

rH 

to 

/< 

00 © 05 CO CO 05 05 05 O Gf COM 

* ^ji t— ^ji co t'- t- o i-h*o 

i-5dcoi-<dco co i-5 d t-i ho 

+ + + + 1 + + + + + + + 


to 

"©• 

-5.35 
-2. 49 
+ 1.53 
+2. 72 
+3.58 

+1.53 

+0.68/ 

-2. 49 

-1.11/ 

-5.35 
-2.39/ 

o 

iH 

»o 

/< 

CO *0 05 

U1 CO 00 CO 

i—5 05 05 © 

11+ + 

o 

»o 

-e- 

-2.72 

-1.22/ 
-4.95 

-2.21/ 

00 

'T 

*© 

CO CO H -5* 

CO iH 1-1 05 # 

oo ci © 

+ + + + 


*©* 

-3.58 

-1.60/ 

-2.09 

-0.93/ 






















































Normal equations 


168 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 




























































Solution of normals 


APPLICATION OP LEAST SQUARES TO TRTANGULATION. 109 



-13.9036 
+ 0.154459 

I - 

lO CO rH t rt) 

CO T+ GO Oi 

rH -t< lO O 

CO 

O © HO 

+ 4“ +1 

12 

Oi CO I'- WO rH 

CO 1- CO CO 00 

‘O 00 00 »o © 

»t« 00 CO © 

need cd o 

00' t- 

+ 1 + +1 

+59. 5951 

- 6.4492 

+ 14.1867 

- 6.1467 

+61.185S 

- 1.145975 

+ 8.3278 

- 1.9580 

-11.0063 

+26.8689 

-12.0299 

+ 10.2025 

— 0.153605: 

+ 3.9182 

- 0.9738 

+ 1.3538 

+ 2.5190 

+ 8.0303 

- 0.9372 

+13.9098 1 

- 0. 777492J 


+0.5960 

-0.006621 

+ 1. 0071 
-0.0276 

+0.9795 
-0.021647 

-0. 4396 

+0. 3810 

+0.0187 

-0.0399 
+0. 000494 

+0. 5294 

+0. 2765 

+0. 3376 

+0.0033 

+ 1.1468 

-0.021479 

-0. 2811 

+0. 0839 

-0. 2619 

-0.0146 

-0. 2255 

-0. 6992 

+0.010527 

-0.1405 

+0. 0417 

+0.0322 

-0. 0014 

+0.1505 

+0.0642 

+0.1467 

-0. 008200 

rH 





-1.4231 

-1. 4231 

+0.021426 

-1.3059 

+0.1307 

-1.1752 

+0.065688 

CO • 

r-H 





+5.0745 

+5.0745 

-0.076400 

+2.9155 

-0. 4662 

+2. 4493 

-0.136904 

?H 





+1.1977 

+1.1977 

-0.018032 

+0.4153 

-0.1100 

+0. 3053 

-0.017065 

H 

H 





+3.5402 

+3.5402 

-0. 053300 

+2. 7178 

-0.3252 

+2.3926 

-0.133735 

10 

+ 0.7642 
- 0.008490 

- 0.6837 

- 0.0354 

- 0.7191 
+ 0.015892 

+ 1.2251 
+ 0.4885 

- 0.0137 

+ 1.6999 

- 0.021046 

- 2.1972 
+ 0.3545 

- 0.2479 

- 0.1424 

- 2.2330 
+ 0.041823 

-24.1325 
+ 0. 1076 
+ 0.1923 
+ 0.6227 
+ 0.4390 

-22. 7709 
+ 0.342829 

- 8. 1327 
+ 0. 0535 

- 0. 0237 

+ 0. 0584 

- 0.2931 

+ 2.0918 

- 6.2458 

+ 0.349111 


+ 4.3305 
0.048109 

+ 0.6543 
- 0.2008 

+ 0.4535 
- 0.010022 

+ 5.4192 
+ 2.7681 
+ 0.0087 

+ 8.1960 
- 0.101473 

1+ 0.2709 

+ 2.0087 
+ 0.1563 
- 0.6868 

1+ 1.7491 

— 0.032760 

-44. 2628 
+ 0.6099 

- 0.1213 
+ 3.0021 

- 0.3439 

-41.1160 
+ 0.619026 

- 6. 7978 
+ 0. 3033 
+ 0. 0149 
+ 0. 2815 
+ 0. 2296 
+ 3. 7770 

- 2.1915 
+ 0.122494 

00 

+1.8229 
-0.020251 

+0.0158 
-0. 0845 

-0. 0687 
+0.001518 

+2.3685 
+1.1652 
-0.0013 

+3. 5324 
-0.043734 

-0. 6153 
+0. 8456 
-0. 0237 
-0. 2960 

-0.0894 
+0. 001674 

-1. 7394 
+0. 2567 
+0. 0184 
+1. 2939 
+0.0176 

-0.1528 
+0. 002300 

+0. 9316 
+0.1277 
-0. 0023 
+0.1213 
-0.0117 
+0. 0140 

+1.1806 
-0.065990 


+2.6686 
-0.029646 

+1.2372 
-0.1237 

+1.1135 
-0. 024608 

+3. 0590 
+1. 7058 
+0. 0213 

+4. 7S61 
-0.059256 

+2.5098 
+1. 2378 
+0. 3838 
-0. 4011 

+3. 7303 
-0.069866 

-7. 0677 
+0. 3758 
-0. 2978 
+ 1. 7531 
-0. 7334 

-5.9700 
+0.089882 

-2. 2687 
+0. 1869 
+0. 0366 
+0.1644 
+0. 4896 
+0. 5484 

-0. 8428 
+0.047109 

o 

- 6.3049 
+ 0.070043 

- 1.7807 
+ 0.2923 

- 1.4884 
+ 0.032893 

+ 1.2849 

- 4.0302 

- 0.0284 

- 2.7737 
+ 0.034341 

- 3.8022 

- 2.9246 

- 0. 5130 
+ 0.2324 

- 7.0074 
+ 0.131245 

+ 6. 2298 

- 0. 8879 
+ 0. 3980 

- 1.0160 
+ 1.3777 

+ 6.1016 
- 0.091863 

+19. 9567 

- 0. 4416 

- 0. 0490 

- 0. 0953 

- 0. 9197 

- 0. 5605 

+17.8906 

5 X 3 

»o 

-12.6766 
+ 0.140828 

+ 11.5126 
+ 0.5877 

+12.1003 
- 0.267415 

—21. 7139 
- 8.1030 
+ 0.2313 

-29.5856 
+ 0.366294 

+ 9. 7277 

- 5.8801 
+ 4.1708 
+ 2.4791 

+ 10. 4975 

- 0.196612 

+84. 3424 

- 1. 7852 

- 3. 2358 
-10. 8370 

- 2.0639 

+66. 4205 

3<f> 3 

1 

2 

3 

4 

5 


-41. 7538 
+ 0.463854 

-17. 5325 
+ 1.9357 

-15.5968 
+ 0.344687 

+33.7556 
-26. 6894 

- 0.2981 

+ 6.7681 

- 0.083795 

+78. 7027 
-19. 3677 

- 5.3760 

- 0.5671 

+53. 3919 
<5X? 

1 

2 

3 

4 

CO 

- 57.5385 
+ 0.639210 

- 3.5323 
+ 2.6674 

- 0.8649 
+ 0.019114 

+117. 5659 

- 36. 7792 

- 0.0165 

+ 80.7702 
d<f> 2 

1 

2 

3 

<N 

+ 4.1730 
- 0.046359 

r~io <N HN 

(M rO Oi 

uid 

+1 + 

rH 

s 

r-H r-H 

o 

+ '* 


























































Solution of normals —Continued 


* 


170 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


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cmooxoooo oo 

I++I++I+ I + 


aoiio-f toc/oo I- <m 

(ZiOO'C'IM'tiocol't- OO 

OOOH<tOHOMH I'IQ 
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05 tP N- 40 

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CM 40 x CO 
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g o O O CM O 
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APPLICATION OF LEAST SQUARES TO TRIANGULATION. 171 


N 

- 4.4519 

- 0.5438 

- 1.8602 
+ 12.4778 

- 1.4472 
+ 3.9690 
+ 5.1624 

+ 13.3061 

- 0.161640 

- 0.2802 

- 0.1840 

- 0.2374 

- 5.4013 

- 0.5414 

- 0.8730 

+ 1.5823 

+ 0.7360 

- 5.1990 

+ 0.204684 

- 3.5317 

- 0.7795 

- 1.9043 

+ 4.8302 

+ 0.1445 

+ 12.7212 

- 7.2024 

+ 3.0893 

- 0.3239 

+ 7.0433 

- 0.140699 

+ 2.1778 

+ 0.2186 

+ 0.9137 

- 1.0863 

- 0.0192 

+ 0.5864 

+ 2.8%8 

+ 1.8413 

- 3.5367 

- 0.1549 

+ 3.7470 

- 0.363486 


-0.5202 
+0.0373 
-0.0196 
-0.0446 
-0.0326 
+0.0647 
+0.0510 

-0. 4640 
+0.005637 

-0.0948 
+0.0126 

-0.0025 

+0.0193 

-0.0122 

-0.0142 

+0.0156 

-0.0257 

-0.1019 

+0.004012 

-0.2161 

+0.0534 

-0.0201 

-0.0172 

+0.0033 

+0.2072 

-0.0711 

-0.1077 

-0.0063 

-0.1746 

+0.003488 

+0.0496 

-0.0150 

+0.00% 

+0.0039 

-0.0004 

+0.00% 

+0.0277 

-0.0642 

-0.0693 

+0.0038 

-0.0447 

+0.004336 


-3.7670 

-3.7670 

+0.045761 

+1.2720 

-0.2084 

+ 1.0636 

-0.041874 

-2.6458 

-0.8746 

+0.0663 

-3.4541 

+0.069000 

-0.9735 

-0.5213 

+0.7235 

+0.0760 

-0.6953 

+0.067449 

H 

<N 

+6.3184 

+6.3184 

-0.076755 

-3.9798 

+0.3495 

-3.6303 
+0.142925 

+7.0946 

• 

+1.4670 

-0.2262 

+8.3354 

-0.166511 

+0.9507 

+0.8743 

-2.46% 

-0.1833 

-0.8279 

+0.080312 

O 

(M 

+1.0673 

+1.0673 

-0.012965 

-1.0111 

+0.0590 

-0.9521 
+0.037484 

+1.6860 

+0.2478 

-0.0593 

+1.8745 

-0.037446 

+0.0354 

+0.1477 

-0.6477 

-0.0412 

-0.5058 

+0.049%6 

O 

rH 

+6.7770 

+6.7770 
-0.082326 

-2.6480 

+0.3748 

-2.2732 

+0.089496 

+5.2774 

+1.5734 
-0.1416 

+6. 7092 
-0.134025 

+1.6185 

+0.9378 

-1.5464 

-0.1476 

+0.8623 

-0.083649 

00 

rH 

+ 1.5610 

- 0.6871 
+ 0.0291 

- 0.3066 

- 0.5219 

+ 0.0745 

- 0.000905 

- 1.9532 

+ 0.2974 
+ 0.0109 
+ 0.0674 

- 0.1600 
+ 0.0041 

- 1.7334 
+ 0.068244 

-11.4363 

- 0.2660 

- 0.0029 

- 0.9828 
+ 0. 7282 
+ 0.0173 

- 0.1080 

-12.0505 
+ 0.240724 

- 3.6212 

+ 0.0598 
+ 0.0004 

- 0.0453 

- 0.2838 
+ 0.0103 

- 1.1792 
+ 0.2651 

- 4.7939 

+ 0.465043 

t- 

rH 

-15.1399 

+ 1.4000 

- 0.4208 
+ 1.1710 

- 0.4871 

-13.4768 
+ 0.163714 

+ 6.3530 

- 0.6060 

- 0.1574 

- 0.2576 

- 0.1493 

- 0.7454 

+ 4.4373 

- 0.1746% 

-32.5324 

+ 0.5420 
+ 0.0420 
+ 3. 7533 
+ 0.6795 
- 3.1290 
+ 0.2764 

-30.3682 
+ 0.606645 

+ 8.1200 

- 0.1219 

- 0.0056 
+ 0.1730 

- 0.2648 

- 1.8649 
+3.0185 
+ 0.6679 

+ 9.7222 

- 0.943125 

o 

rH 

+24.4953 

- 0.0728 

- 0.1377 
+ 0.1802 

- 0.6591 

+23.8059 

- 0.289190 

- 6.7932 

+ 0.0315 

- 0.0515 

- 0.03% 

- 0.2020 
+ 1.3168 

- 5.7380 
+ 0.225905 

+ 0.4952 

- 0.0282 
+ 0.0138 
+ 0.5776 
+ 0.9195 
+ 5.5271 

- 0.3575 

+ 7.1475 

- 0.142781 

- 0.3391 

+ 0.0063 

- 0.0018 
+ 0.0266 

- 0.3583 
+ 3.2942 

- 3.9034 

- 0.1572 

- 1.4327 
+ 0.138982 

lO 

rH 

-57.7023 

+ 2.0459 
- 0.3555 
+ 1.3193 
+ 0.4010 

-54.2916 
+ 0.659525 

+ 1.3790 

- 0.8S56 

- 0.1330 

- 0.2902 
+ 0.1229 

- 3.0030 

- 2.8099 
+ 0.11%26 

-13.8524 

+ 0.7920 
+ 0.0355 
+ 4.2284 

— 0.5595 
-12.6052 

- 0.1751 

-22.1363 
+ 0.442202 

- 0.1385 

- 0.1781 

- 0.0047 
+ 0.1949 
+ 0.2180 

- 7.5127 

- 1.9115 
+ 0.4869 

- 8.8457 
+ 0.85S098 

-r 

rH 

- 8.8081 
+ 0.0759 
+ 0.1572 
+ 0.8498 

- 0.0243 

- 0.4437 

- 3.1979 

-11.3911 
+ 0.138377 

-15.4350 
+ 0.0257 
+ 0.0201 

- 0.3678 

- 0.0091 
+ 0.0976 

- 0.9802 

- 0.6301 

-17.2788 
+ 0.680265 

+ 1.1818 
+ 0.1087 
+ 0.1609 
+ 0.3289 
+ 0.0024 

- 1.4220 
+ 4.4616 

- 2.6447 

- 1.0765 

+ 1.1011 

- 0.021995 

+25.6494 

- 0.0305 

- 0.0772 

- 0.0740 

- 0.0003 

- 0.0655 

- 1.7387 

- 1.5763 
-11.7542 

- 0.0242 

+10.3085 

CO 

rH 

-13.4998 

- 0.2705 

- 0.3276 

- 3.7784 
+ 0.1826 

- 9.6249 
+ 8.2061 

-19.1125 
+ 0.232175 

- 6. 7278 

- 0.0915 

- 0.0418 
+ 1.6356 
+ 0. %83 
+ 2.1169 
+ 2.5151 

- 1.0572 

- 1.5824 
+ 0. %2299 

iOONO)OOH(NCO 

rH rH rH rH 

cocp(Moooo^o co 

CO «Q C'l 00 00 X N C/j O r- 

OGOcO'OrH-r-rcoOi -a. 

O CO CO n-o X rr-r O O 

Oi ©0*H00rH-r0 o 

CO rH to 

+ 11 II 1 1 1 1 + 

M 

rH 

- 6.8476 

- 0.0038 

- 0.0408 
+ 4.2252 

- 0.6839 
+ 0.6605 

- 1.8028 

- 4.5532 
+ 0.055312 

+28. 4613 
- 0.0216 

- 0.0052 

- 1.8290 

- 0.2558 

- 0.1453 

- 0.5525 

- 0.2518 

+25.4001 

o'X 6 

5 

6 

t 

8 

9 

10 

11 

12 

rH 

rH 

+ 103.3016 

- 0.1887 

- 0.3200 

- 9.7608 

- 1.8281 

- 3.0030 

- 5.8818 

+ S2.3192 
3</>g 

5 

6 

7 

8 

9 

10 

11 

* 

iO *>0 N 00 Ci O 

H 















































































































Solution of normals —Continued 


172 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


173 


Solution of normals —Continued 



18 

19 

20 

21 

22 

V 



+ 

61.5179 

_ 

1.0383 

- 2.8738 

+ 

31.3248 


8.1976 

+ 

7. 4331 

+ 7.1924 

7 

— 

0.0484 








— 

0.0031 

+ 0.8784 

8 

— 

0.0005 








+ 

0.0005 

+ 0.0230 

9 

— 

0.0313 








+ 

0.0066 

+ 0.4053 

10 

— 

0.0463 








+ 

0.0045 

+ 0.4581 

11 

— 

0.0001 

— 

0.0061 

- 0.0010 

— 

0.0057 

+ 

0.0034 

+ 

0.0004 

- 0.0120 

12 

— 

0.1183 

— 

0.1551 

- 0.0650 

— 

0.2477 

+ 

0.0726 

— 

0. 0070 

- 0.3548 

13 

— 

2.9008 

+ 

1. 6151 

+ 0. 4512 

+ 

2.0065 ' 

— 

0.8315 

— 

0.0420 

+ 1.6955 

14 

— 

2.2294 

+ 

0. 4010 

- 0.2352 

— 

0.3850 

— 

0. 3233 

— 

0.0208 

+ 1. 7425 

15 

— 

0.5441 

— 

3.2335 

+ 0.9685 

— 

0. 8120 

+ 

0.2910 

+ 

3.2905 

+ 3.4243 

16 

— 

35.3839 

— 

14.8119 

- 4.3115 

— 

0.9158 

— 

3.6707 

— 

7. 2125 

- 7.9221 

17 

— 

1.3061 

— 

1.8508 

+ 0.8318 

— 

10.5336 

+ 

0.3480 

+ 

7.0757 

+ 5.7131 


+ 

18.90S7 

_ 

19.0796 

- 5.2350 

+ 

20. 4315 

_ 

12.3081 

+ 

10.5259 

+ 13.2434 



<JX 9 

+ 

1.009038 ) 

+ 0.276857 

— 

1.080534 

+ 

0.650923 

— 

0.556670 

- 0.700387 




+ 106. 8200 

+27.9S99 

_ 

16.8102 

_ 

7.0763 

+ 

34. 6511 

+ 85.3611 



11 

— 

0. 5579 

- 0.0879 

— 

0.5202 

+ 

0. 3101 

+ 

0.0382 

- 1.0954 



12 

— 

0.2034 

- 0.0852 

— 

0.3249 

+ 

0.0952 

— 

0.0091 

- 0.4653 



13 

— 

0.8992 

- 0.2512 

— 

1.1172 

+ 

0. 4629 

+ 

0. 0234 

- 0.9440 



14 

— 

0.0721 

+ 0.0423 

+ 

0.0693 

+ 

0.0582 

+ 

0.0037 

- 0.3134 



15 

— 

19.2169 

‘+ 5.7557 

— 

4.8257 

+ 

1.72% 

+ 

19. 5557 

+ 20.3509 



16 

— 

6.2003 

- 1.8048 

— 

0.3834 

— 

1.5366 

— 

3.0192 

- 3.3162 



17 

— 

2.6226 

+ 1.1786 

— 

14.9264 

+ 

0. 4931 

+ 

10.0264 

+ 8.0956 



18 

— 

19.2520 

- 5.2S23 

+ 

20.6162 

— 

12.4193 

+ 

10.6210 

+ 13.3631 




+ 

57. 7956 

+27.4551 

_ 

18.2225 

_ 

17.8831 

+ 

71. 8912 

+121.0363 




d < f>io 

- 0.475038 

+ 

0.315292 

+ 

0.309420 

— 

1.243887 

- 2.094213 






+50. 9179 

__ 

9. 2141 

_ 

26.1027 

+ 167.6744 

+214. 2334 





11 

- 0.0138 

— 

0.0819 

+ 

0.0488 

+ 

0.0060 

- 0.1725 





12 

- 0.0357 

— 

0.1361 

+ 

0.0399 

— 

0.0038 

- 0.1949 





13 

- 0. 0702 

— 

0.3121 

+ 

0.1293 

+ 

0.0065 

- 0.2637 





14 

- 0.0248 

— 

0.0406 

— 

0.0341 

— 

0.0022 

+ 0.1839 





15 

— 1.7239 

+ 

1. 4454 

— 

0.5180 

— 

5.8571 - 

- 6.0953 





16 

- 0.5253 


0.1116 

— 

0. 4473 

— 

0.8788 

- 0. %53 





17 

- 0.5297 

+ 

6. 7081 

— 

0. 2216 

— 

4.5060 

- 3.6382 





18 

- 1.4493 

+ 

5.6566 

— 

3. 4076 

+ 

2.9142 

+ 3.6665 





19 

-13.0422 

+ 

8.6564 

+ 

8.4952 

— 

34.1511 

- 57.4%8 






+33.5030 

+ 

12. 5701 

_ 

22.0181 

+125.2021 

+149.2571 





• 

<JXio 


0. 375193 

+ 

0.657198 

— 

3. 737041 

- 4.455037 







+ 173.0271 

_ 

35. 3120 

-162.3283 

-115. 8122 






11 

— 

0. 4850 

+ 

0.2891 

+ 

0.0356 

- 1.0213 






12 

_ 

0. 5189 

+ 

0.1520 

— 

0. 0146 

- 0.7431 






13 

_ 

1.3879 

+ 

0. 5751 

+ 

0.0291 

- 1.1728 






14 

_ 

0.0665 

— 

0.0558 

— 

0.0036 

+ 0.3009 






15 

_ 

1. 2118 

+ 

0. 4343 

+ 

4.9108 

+ 5.1105 






16 

_ 

0.0237 

— 

0.0950 

— 

0.1867 

- 0.2050 






17 

_ 

84.9517 

+ 

2.8063 

+ 57.0642 

+ 46.0750 






18 

_ 

22. 0769 

+ 

13. 2993 

— 

11. 3736 

- 14.3099 






19 

_ 

5. 7454 

— 

5.6384 

+ 

22.6667 

+ 38.1618 






20 

- 

4. 7162 

+ 

8. 2610 

— 

46.9750 

- 56.0002 







+ 

51. 8431 

_ 

15.2841 


136.1754 

- 99.6164 







d < f > it 

+ 

0. 294815 

+ 

2.626683 

+ 1.921498 









+ 

54. 7848 

+188.1235 

+163.8779 








11 


0.1724 

— 

0. 0212 

+ 0.6089 








12 

— 

0.0445 

+ 

0. 0043 

+ 0.2177 








13 

— 

0. 2383 

— 

0.0120 

+ 0.4860 








14 

— 

0.0469 

— 

0.0030 

+ 0.2527 








15 

— 

0.1557 

— 

1.7601 

- 1.8317 








16 

_ 

0.3808 

— 

0. 7482 

- 0.8218 








17 

— 

0.0927 

— 

1. 8851 

- 1.5220 








18 

— 

8.0116 

+ 

6. 8516 

+ 8.6204 








19 

— 

5. 5334 

+ 

22. 2446 

+ 37.4511 








20 

— 

14. 4703 

+ 

82.2826 

+ 98.0915 








21 

— 

4. 5060 

— 

40.1466 

- 29.36S4 









+ 

21.1322 

+254.9304 

+276.0626 









iXu 

1 " 

12.06360 

- 13.06360 







































174 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Bach solution 


22 


-12.0636 


-12.0636 


21 


+2.6267 

-3.5565 


-0.9298 


20 


- 3.7370 

- 7.9282 
+ 0.3489 


-11.3163 


19 


-1.2439 
-3. 7327 
-0.2932 
+5.3757 


+0.1059 


18 


- 0.5567 

- 7.8525 
+ 1.0047 

- 3.1330 
+ 0.1069 


-10.4306 


17 


-0.6347 
+0.3765 
-0.8785 
+0.8443 
+0.0176 
- 1.2220 


-1.4968 


16 


+0.1657 
-1.0174 
-0.0196 
- 1.1210 
+0.0360 
-8.4798 
+0.5110 


-9.9251 


15 


-0.5425 
+0.5788 
-0.1245 
+ 1.8069 
+0.0565 
-0.9356 
-0.5808 
-0.4057 


-0.1469 










14 

13 

12 

11 

10 

9 

8 

7 

+0.0043 
-0.8137 
-0.0747 
-0.5552 
-0.0089 
-4.8507 
+ 1.4117 
-1.3794 
-0.1261 

+0.0035 
-0.8324 
+0.1548 
+0.4238 
-0.0142 
-2.5109 
-0.9080 
+ 1.4171 
-0.0650 
+0.1406 

+0.0040 
+0.5052 
-0.1329 
-0.4242 
+0.0095 
-0.7118 
+0.2615 
-2.2421 
-0.0163 
-4.3487 
-0.1365 

+0.0056 
-0.5520 
+0.0714 
+0.1467 
-0.0087 
+0.0094 
-0.2450 
+2.8702 
-0.0969 
-0.8846 
-0.5086 
-0.4000 

-0.0056 
-0.6012 
-0.0805 
-0.7224 
+0.0065 
-2.2577 
+ 1.9853 
-1.4399 
+0.2647 

-0.0042 
-0.2065 
+0.1132 
+0.1155 
+0.0125 
-O'. 1831 
-1.3614 
+0.3084 
+0.0790 
-0.2828 

+0.0041 
+0.0385 
-0.0799 
-0.1733 
-0.0066 
-0.0197 
+0.0507 
-0.6273 
+0.0945 
-2.2905 
+0.0490 

+0.0019 
-0.2996 
+0.0876 
-0.0302 
+0.0126 
+0.2271 
-0.3460 
+ 1.2773 
+0.1663 
-0.0681 
-0.2829 
-0.0309 

-6.3927 

-2.8508 

-2.1907 

-1.4094 


-7.2323 


-2.9605 



+0.4075 



+0. 7151 

6 

5 

4 

3 

2 

1 



-0.0082 
-0.4199 
+0.2999 
+0.1234 
-0.0545 
-0.9952 
-0.1726 
+0.1954 
+0.0337 

+0.0105 
-0.1370 
+0.1674 
+0.1304 
-0 0217 
-0.9773 
-0.8725 
* -0.0068 
+0.0643 
+0.0917 

-0.0215 
-0.1192 
+0.0462 
-0.0050 
-0.0500 
-0.1310 
+0.3049 

+0.0005 
+0.0600 
+0.1430 
+0.1295 
-0.0424 
-0.0343 
-0.5681 
-0.0020 

-0.0216 

-0.0453 

+0.0141 

-0.0045 

-0.0176 

-0.0328 

+0.4148 

+0.0084 

-0.0060 

-0.0066 
+0.0242 
+0.0678 
+0.0600 
-0.0212 
-0.0699- 
-0.2184 
+0.0113 
-0.2006 
-0.0143 



+0.0244 



-0.3138 



-0.9980 


+0.3095 



-1.5510 



-0.3677 







































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 175 


Computation of corrections. 


1 

2 

2 l 

3 

4 

5 

3 a 

Zj 

+ 0.700 

+ 0.7 

- 0.842 

- 0.857 

+ 0.700 

+ 0.3 

- 0.842 
- 0.857 
+ 0.3 

- 1.399 

+ 0.700 

+ 1.039 
+ 1.0 

- 1.318 
- 0.098 
+ 1.039 
- 0.6 

+ 0.301 

- 1.541 

+ 1.039 

- 0.9 

+ 1.039 
+ 1.0 

+ 0.301 

- 1.541 

- 1.318 

- 0.098 

- 1.5 

- 4.156 
+ 1.039 


- 0.699 

- 0.7 

- 0.977 

- 1.0 

- 1.101 

- 1.1 

4 a 

5 a 

6 

7 

8 

Z3 

17 

15 

- 0.842 
- 0.857 
+ 1.592 

+ 0.301 
- 1.541 
+ 1.592 

+ 0.978 
- 0.699 
+ 1.592 
- 0.9 

+ 0.971 

+ 1.0 

+ 2.416 
- 0.699 
- 2.062 
+ 0.055 
+ 1.592 
+ 1.4 

+ 1.254 
+ 0.149 
- 5.289 
+ 0.479 
+ 1.592 
- 2.1 

+ 4.107 
- 3.649 
- 2.062 
+ 0.055 
- 5.289 
+ 0.479 
- 1.6 

- 7.959 
+ 1.592 

+ 0.857 
- 0.131 
+ 1.860 

+ 2.416 
- 0.699 
- 2.062 
+ 0.055 
+ 1.860 
- 2.7 

- 0.107 
- 0.1 

. + 0.352 

+ 0.4 

+ 2.586 

+ 2.6 

+ 2.702 

+ 2.7 

- 3.915 

- 3.9 

- 1.130 

- 1.1 

16 

24 

14 

9 

10 

11 

12 

13 

- 1.318 
- 0.098 
+ 1.860 
- 1.9 

+ 2.416 
- 0.699 
- 2.523 
- 0.174 
- 4.6 

- 5.580 

+ 1.860 

- 0.753 

- 0.8 

+ 2.624 

- 1.836 

- 0.753 

+ 0.4 

- 4.045 
+ 3 . 792 
- 0.753 
+ 0.7 

- 2.295 
+ 3.174 
- 0 . 753 
+ 1.8 

+ 0.857 
- 0.131 
- 0.753 
- 0.6 

+ 0.978 
- 0.699 
- 0 . 753 
- 0.2 

- 1.456 

- 1.4 

+ 0.435 

+ 0.4 

- 0.306 
- 0.3 

+ 1.926 

+ 1.9 

- 0.627 

- 0.7 

- 0.674 
- 0.7 

25 

18 

19 

20 

21 

22 

Z 6 


+ 0.978 
- 0.699 
+ 0.857 
- 0.131 
- 2.295 
+3 174 
+ 2.624 
- 1.836 
- 4.045 
+ 3.792 
+ 2.1 

+ 1.254 
+ 0.149 
- 5.289 
+ 0.479 
+ 3.728 

+ 0.321 

+ 0.3 

+ 1.440 
+ 0.062 
- 7.119 
+ 2.545 
+ 3.728 
- 1.8 

- 2.295 
+ 3.174 
+ 3 . 728 
- 2.8 

+ 1.807 
+ 1.8 

+ 2.373 
+ 2.275 
+ 1.094 
- 6.750 
+ 3 . 728 
- 2.9 

+ 6.980 
+ 1.976 
- 6.342 
- 5.645 
+ 3 . 728 
- 1.5 

+ 1.254 
+ 0.149 
+ 1.440 
+ 0.062 

- 5.351 
+ 10.449 
+ 1.094 

- 6 . 750 

- 6.342 

- 5.645 

- 9.0 

- 1.144 

- 1.2 

- 0.180 

- 0.2 

- 0.803 

- 0.8 

+ 4.519 
- 0.753 

- 18.640 
+ 3.728 

28 

29 

30 

31 

32 

Z7 


- 2.560 
- 1.066 
+ 1.454 
+ 2.604 
- 0.633 

- 0.201 

- 0.1 

- 1.495 

- 6.247 

- 4.579 
+ 13.489 

- 0.633 
+ 0.3 

- 0.179 
- 13.085 

- 0.352 
+ 12.601 

- 0.633 
+ 0.9 

+ 2.373 
+ 2.275 
+ 1.094 
- 6 . 750 
- 0.633 
+ 0.6 

+ 2.624 
- 1.836 
- 0.633 
+ 1.0 

+ 1.155 
+ 1.2 

+ 2.373 
+ 2.275 

- 0.515 
- 28.983 

- 0.352 
+ 12.601 
+ 1.454 
+ 2.604 

- 4.579 
+ 13.489 
+ 2.8 

+ 3.167 

- 0.633 

+ 0.835 
+ 0.9 

- 0 . 748 

- 0.7 

- 1.041 

- 1.0 


91865°—15-12 




























































































176 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Computation of corrections —Continued 


23 

24 

25 

26 

27 

28 

+ 6.980 
+ 1.976 
- 6.342 
- 5.645 
+ 1.588 

- 4.045 
+ 3.792 
+ 1.588 
- 1.4 

- 0.179 
- 13.085 

- 0.352 
+ 12.601 
+ 1.588 

- 0.6 

+ 3.834 
+ 4.162 
+ 1.108 
- 10.559 
+ 1.588 
- 0.3 

+ 6.977 
- 2.480 
- 10.844 
+ 5.562 
+ 1.588 
+ 0.9 

+ 6.980 
+ 1.976 

- 0.179 
- 13.085 
+ 0.070 
+ 12.429 
+ 1.108 
- 10.559 
- 10.844 
+ 5.562 

- 1.4 

- 0.065 

- 0.0 

- 1.443 

- 1.4 

- 0.027 

- 0.0 

- 0.167 

- 0.2 

+ 1.703 
+ 1.7 






- 7.942 
+ 1.588 

38 

39 

40 

41 

42 

29 

- 2.180 

- 8.534 

- 0.786 
+ 11.712 

- 1.284 

- 1.015 
- 17.358 

- 3.727 
+ 25.033 

- 1.284 
+ 3.4 

+ 0.623 
- 27.410 
+ 3.352 
+ 24.228 

- 1.284 

- 2.2 

+ 3.834 
+ 4.162 
+ 1.108 
- 10.559 
- 1.284 
+ 1.6 

- 2.560 

- 1.066 
+ 1.459 
+ 2.604 
- 1.284 
+ 0.7 

- 2.560 

- 1.066 
+ 3.834 
+ 4.162 

- 0.004 
- 61.258 
+ 3.352 
+ 24.228 

- 0.786 
+ 11.712 

- 3.727 
+ 25.033 
+ 3.5 

- 1.072 

- 1.1 

+ 5.049 
+ 5.0 

- 2.691 

- 2.7 

- 1.139 

- 1.2 

- 0.147 

- 0.1 






+ 6.420 
- 1.284 

33 

34 

35 

36 

37 

210 

+ 6.977 

- 2.480 
- 10.844 
+ 5.562 

- 2.538 

- 1.495 

- 6.247 

- 4.579 
+ 13.480 

- 2.538 
+ 3.0 

+ 0.623 
- 27.410 
+ 3.352 
+ 24.228 
- 2.538 
+ 3.5 

+ 6.265 
+ 10.228 

- 0.420 
- 15.880 

- 2.538 
+ 2.2 

+ 8.719 
+ 0.384 
- 5.957 
- 0.626 
- 2.538 
+ 0.1 

- 1.495 

- 6.247 
+ 6.977 

- 2.480 
+ 0.623 
- 27.410 
+ 2.914 
+ 53.890 

- 0.420 
- 15.880 

- 5.957 

- 0.626 
+ 8.8 

- 3.323 

- 3.3 

+ 1.630 
+ 1.7 

+ 1.755 
+ 1.8 

- 0.145 

- 0.1 

+ 0.082 

+ 0.1 






+ 12.689 
- 2.538 

43 

44 

45 

46 

47 

2 ll 

+ 8.719 
+ 0.384 
- 5 . 957 
- 0 . 626 
- 2 . 716 

- 1.015 
- 17.358 

- 3.727 
+ 25.033 

- 2.716 

- 1.8 

+ 0.025 
- 55.481 

- 0.254 
+ 58.307 

- 2.716 

- 0.6 

+ 4.655 
+ 22.009 
+ 0.329 
- 23.877 

- 2.716 

- 2.0 

+ 10.478 
- 12.412 

- 6.509 
+ 14.356 

- 2.716 
+ 0.9 

- 1.015 
- 17.358 
+ 8.719 
+ 0.384 
+ 0.025 
- 55 . 481 
+ 5.194 
+ 92.311 
+ 0.329 
- 23.877 

- 6.509 
+ 14.356 

- 3.5 

- 0.196 
- 0.2 

- 1.583 

- 1.6 

- 0.719 

- 0.7 

- 1.600 
- 1.6 

+ 4.097 
+ 4.1 






+ 13.578 
- 2.716 















































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


177 


Computation of corrections —Continued 


48 

49 

50 

51 

52 

Zl 2 

+ 0.720 
+ 0.199 
+ 0.519 
- 0.226 
- 1.196 

+ 0.391 
- 29.478 

- 2 . 473 
+ 35.829 

- 1.196 

- 2.3 

+ 0.025 
- 55.481 

- 0.254 
+ 58.308 

- 1.196 

- 0.3 

+ 6.265 
+ 10.228 

- 0.420 
- 15.880 

- 1.196 
+ 0.8 

- 2.180 

- 8.534 

- 0.786 
+ 11.712 

- 1.196 

- 0.7 

- 2.180 

- 8.534 
+ 6.265 
+ 10.228 

- 0.071 
- 88.929 

+ 0.016 

+ 0.0 

+ 0 . 773 
+ 0.8 

+ 1.102 
+ 1.1 

- 0.203 

- 0.2 

- 1.684 

- 1.7 

— U . ZD 4 

+ 58.308 
+ 0.519 

- 0.226 

- 2.473 
+ 35.829 

- 2.5 






+ 5.982 
- 1.196 

53 

54 

55 

56 

57 

Zl3 

+ 10 . 478 
- 12.412 

- 6.509 
+ 14.356 

- 7.305 

+ 0.391 
- 29.478 

- 2 . 473 
+ 35.829 

- 7.305 
+ 0.7 

+ 0.177 
- 48.094 
+ 1.553 
+ 51.270 

- 7.305 

- 1.8 

+ 1.832 
+ 22.318 
- 7.305 
- 11.0 

+ 5.607 
- 29.918 
- 7.305 
+ 33.7 

+ 0.391 
- 29.478 
+ 10 . 478 
- 12.412 
+ 0.177 
- 48.094 
+ 0.009 
+ 93.855 
+ 21.6 

+ 5.845 
+ 5.8 

+ 2.084 
+ 2.1 

- 1.394 

- 1.4 

- 2.336 

- 2.3 

- 4.199 

- 4.2 






+ 36.526 
- 7.305 

58 

59 

60 

61 

62 

zu 

- 0.530 
+ 20.030 

- 4.333 
- 15.2 

- 0.116 
- 36.552 
- 4.333 
+ 43.2 

+ 0.177 
- 48.094 
+ 1.553 
+ 51.270 
- 4.333 
+ 1.3 

+ 4 . 655 
+ 22.009 
+ 0.329 
- 23.877 

- 4.333 

- 0.5 

+ 0.720 
+ 0.199 
+ 0.519 
- 0.226 
- 4.333 
+ 0.8 

+ 0.720 
+ 0.199 
+ 4.655 
+ 22.009 
+ 0.379 
- 88 . 720 
+ 1.553 
+ 51.270 
+ 29.6 

- 0.033 

- 0.0 

+ 2.199 
+ 2.2 

+ 1.873 
+ 1.9 

- 1.717 

- 1.7 

- 2.321 
- 2.3 



* 



+ 21.665 
- 4.333 


63 


+ 5.607 
- 29.918 
- 10.240 
+ 33.2 


- 1.351 

- 1.3 


64 


- 0.116 
- 36.552 
- 10.240 
+ 46.9 


- 0.008 

- 0.0 


65 


- 10.240 

+ 11.6 


+ 1.360 
+ 1.3 


Z15 


- 0.116 
- 36.552 
+ 5.607 
- 29.918 
+ 91.7 


+ 30 . 721 
- 10.240 


66 


- 9.016 
+ 13.3 


+ 4.284 
+ 4.2 


67 


+ 1.832 
+ 22.318 
- 9.016 
- 13.8 


+ 1.334 
+ 1.3 


68 


- 0.530 
+ 20.030 

- 9.016 
- 16.1 


- 5.616 
- 5.6 


Zl6 


- 0.530 
+ 20.030 
+ 1.832 
+ 22.318 
- 16.6 


+ 27.050 
- 9.016 



























































178 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Final computation of triangles 


Symbol 

Station 

Observed 

angle 

Cor¬ 

rec¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Loga¬ 

rithm 



O 1 II 

it 

it 

it 

o / a 

4.050203 

0.158967 

9.830179 

9.999173 

—4a+5a 
-1+2 
+3a— 5 

Fort Morgan-Dauphin Island 
east base 

Cedar 

Fort Morgan 

Dauphin Island east base 

43 54 21.8 

42 33 37. 7 
93 31 59. 6 

+0.5 

-1.4 

+2.1 

22.3 

36.3 
61.7 

0.1 

0.1 

0.1 

22.2 

36.2 

32 01.6 


Cedar-Dauphin Island east base 
Cedar-Fort Morgan 


+1.2 


0.3 


4.039349 

4.208343 

—5a+ 6 
— 3+ 5 
-13+14 

Dauphin Island east base-Dau- 
phin Island west base 

Cedar 

Dauphin Island east base 
Dauphin Island west base 

37 26 33.0 
103 55 36. 4 

38 37 52. 5 

+0.6 

-2.1 

-0.1 

33.6 

34.3 

52.4 

0.1 

0.1 

0.1 

33.5 

34.2 

52.3 

4 . 027832 

0.216120 

9. 987043 

9. 795397 


Cedar-Dauphin Island west base 
Cedar-Dauphin Island east base 


-1.6 


0.3 


4.230995 

4.039349 

-15+16 
—5a+ 7 
-4+5 

Cedar-Dauphin Island east base 
Cat 

Cedar 

Dauphin Island east base 

69 30 32. 7 
60 11 14.2 
50 18 11. 4 

-0.3 

+2.3 

-0.1 

32.4 

16.5 
11.3 

0.0 

0.1 

0.1 

32.4 

16.4 
11.2 

4.039349 

0.028387 

9.938350 

9.886171 


Cat-Dauphin Island east base 
Cat-Cedar 


+1.9 


0.2 


4.006086 

3.953907 

-15+17 
— 6+7 
-12+13 

Cedar-Dauphin Island west base 
Cat 

Cedar 

Dauphin Island west base 

135 31 54.9 
22 44 41.2 
21 43 18. 7 

+3.7 

+1.7 

0.0 

58.6 
42.9 

18.7 

0.1 

0.0 

0.1 

58.5 
42.9 

18.6 

4.230995 

0.154592 

9. 587301 

9.568320 


Cat-Dauphin Island west base 
Cat-Cedar 


+5.4 


0.2 


3.972888 

3.953907 

-16+17 

-3+4 

-12+14 

Dauphin Island east base-Dau- 
phin Island west base 

Cat 

Dauphin Island east base 
Dauphin Island west base 

66 01 22.2 
53 37 25.0 
60 21 11.2 

+4.0 

-2.0 

—0.1 

26.2 

23.0 

11.1 

0.1 

0.1 

0.1 

26.1 

22.9 

11.0 

4 . 027832 

0.039189 

9.905867 

9.939065 


Cat-Dauphin Island west base 
Cat-Dauphin Island east base 


+1.9 


0.3 


3.972888 

4.006086 

-18+19 

-7+8 

Cedar-Cat 

Pins 

Cedar 

Cat 

23 22 40. 3 
30 54 61.5 
18.4 

-1.5 

-6.6 

38.8 

54.9 
26.5 

0.1 

0.1 

0.0 

38.7 

54.8 
125 42 26. 5 

3.953907 

0. 401443 

9. 710768 

9.909561 


Pins-Cat 

Pins-Cedar 




0.2 


4.066118 

4.264911 

-18+20 

-6+8 

-11+13 

Cedar-Dauphin Island west base 
Pins 

Cedar 

Dauphin Island west base 

58 45 26. 0 
53 39 42. 7 
67 34 57.9 

+ 1.5 
-4.9 
-2.6 

27.5 

37.8 

55.3 

0.2 

0.2 

0.2 

27.3 

37.6 

55.1 

4.230995 

0.068044 

9.906076 

9.965872 


Pins-Dauphin Island west base 
Pins-Cedar 


-6.0 


0.6 


4.205115 

4.264911 

-19+20 

-11+12 

Cat-Dauphin Island west base 
Pins 

Cat 

Dauphin Island west base 

35 22 45. 7 
35.3 
45 51 39.2 

+3.0 

-2.6 

48.7 

34.9 

36.6 

o.i 

0.0 

0.1 

48.6 
98 45 34.9 
36.5 

3.972888 

0.237322 

9.994905 

9. 855908 


Pins-Dauphin Island west base 
Pins-Cat 




0.2 


4.205115 

4.066118 























































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 179 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Cor¬ 

rec¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Loga¬ 

rithm 



o t n 

n 

ft 

// 

O / // 



Pins-Dauphin Island west base 






4.205115 

-23+24 

Grand 

54 52 01.6 

+1.4 

03.0 

0.2 

02.8 

0.087341 

-20+22 

Pins 

85 13 07.0 

-2.6 

04.4 

0.1 

04.3 

9.998486 

-10+11 

Dauphin Island west base 

39 54 50.9 

+2.2 

53.1 

0.2 

52.9 

9.807296 




+1.0 


0.5 




Grand-Dauphin Island west base 






4.290942 


Grand-Pins 






4.099752 


Grand-Pins 






4.099752 

-30+31 

Petit 

33 09 08.7 

-0.3 

08.4 

0.1 

08.3 

0.262119 

-23+25 

Grand 

114 03 53.6 

+ 1.4 

55.0 

0.1 

54.9 

9.960510 

-21+22 

Pins 

32 46 57.5 

-0.6 

56.9 

0.1 

56.8 

9.733559 




+0.5 


0.3 




Petit-Pins 






4.322381 


Petit-Grand 






4.095430-' 


Grand-Dauphin Island west base 






4.290942 

-30+32 

Petit 

81 41 28.2 

+1.9 

30.1 

0.1 

30.0 

0.004582 

-24+25 

Grand 

59 11 52.0 

0.0 

52.0 

0.2 

51.8 

9.933962 

- 9+10 

Dauphin Island west base 

39 06 39.1 

-0.7 

38.4 

0.2 

38.2 

9.799905 




+1.2 


0.5 




Petit-Dauphin Island west base 






4.229486 


Petit-Grand 






4.095429 


Pins-Dauphin Island west base 






4.205115 

-31+32 

Petit 

48 32 19.5 

+2.2 

21.7 

0.2 

21.5 

0.125281 

-20+21 

Pins 

52 26 09.5 

-2.0 

07.5 

0.2 

07.3 

9.899090 

- 9+11 

Dauphin Island west base 

79 01 30.0 

+1.5 

31.5 

0.3 

31.2 

9.991984 




+1.7 


0.7 




Petit-Dauphin Island west base 






4.229486 


Petit-Pins 






4.322380+ 1 


Grand-Petit 






4.095429 

-33+34 

Pascagoula 

37 39 20.6 

+5.0 

25.6 

0.1 

25.5 

0.214006 

-25+27 

Grand 

104 18 56.1 

+ 1.7 

57.8 

0.2 

57.6 

9.986300 

-29+30 

Petit 

38 01 38.6 

-1.6 

37.0 

0.1 

36.9 

9.789603 




+5.1 


0.4 




Pascagoula-Petit 






4.295735 


Pascagoula-Grand 






4.099038 


Pascagoula-Grand 






4.099038 

-40+41 

Horn 

38 49 39.0 

+ 1.5 

40.5 

0.1 

40.4 

0.202744 

-33+35 

Pascagoula 

97 55 54.0 

+5.1 

59.1 

0.2 

58.9 

9.995824 

-26+27 

Grand 

43 14 18.9 

+ 1.9 

20.8 

0.1 

20.7 

9.835719 




+8.5 


0.4 




Horn-Grand 






4.297606 


Horn-Pascagoula 






4.137501 


Pascagoula-Petit 






4.295735 

-40+42 

Horn 

77 06 13.2 

+2.6 

15.8 

0.2 

15.6 

0.011094 

-34+35 

Pascagoula 

60 16 33.4 

+0.1 

33.5 

0.2 

33.3 

9.938731 

-28+29 

Petit 

42 37 10.3 

+1.0 

11.3 

0.2 

11.1 

9.830672 




+3.7 


0.6 




Horn-Petit 






4.245560 


Horn-Pascagoula • 






4.137501 


Grand-Petit 






4.095429 

-41+42 

Horn 

38 16 34.2 

+ 1.1 

35.3 

0.2 

35.1 

0.207989 

-25+26 

Grand 

61 04 37.2 

-0.2 

37.0 

0.2 

36.8 

9.942142 

-28+30 

Petit 

80 38 48.9 

-0.6 

48.3 

0.2 

48.1 

9.994187 




+0.3 


0.6 




Horn-Petit 






4.245560 


Horn-Grand 






4.297605+ 1 


































180 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Cor¬ 

rec¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Loga¬ 

rithm 



orn 

it 

it 

it 

o ttt 



Pascagoula-H orn 






4.137501 

—43+44 

Belle “ 

48 58 49. 8 

— 1.4 

48.4 

0.1 

48.3 

0.122351 

-35+37 

Pascagoula 

69 43 28.0 

-1.7 

26.3 

0.2 

26.1 

9. 972219 

-39+40 

Horn 

61 17 53. 5 

-7.7 

45.8 

0.2 

45.6 

9.943055 




-10.8 


0.5 




Belle-IIorn 






4.232071 


Belle-Pascagoula 






4.202907 


B elle-Pascagoula 






4.202907 

-50+51 

Club 

58 40 43. 3 

-1.3 

42.0 

0.1 

41.9 

0. 068410 

—43+45 

Belle 

89 28 55. 3 

-0.5 

54.8 

0.2 

54.6 

9. 999982 

-36+37 

Pascagoula 

31 50 23. 4 

+ 0.2 

23.6 

0.1 

23.5 

9. 722261 




-1.6 


0.4 




Club-Pascagoula 






4.271299 


Club-Belle 






3.9935 1 8 


Belle-Horn 






4.232071 

-50+52 

Club 

105 43 56.9 

-2.8 

54.1 

0.1 

54.0 

0.016580 

—44+45 

Belle 

40 30 05. 5 

+ 0.9 

06.4 

0.1 

06.3 

9.812560 

-38+39 

Horn 

33 45 53. 7 

+ 6.1 

59.8 

0.1 

59.7 

9. 744927 




+ 4.2 


0.3 




Club-Horn 






4.061211 


Club-Belle 






3.993578 


Pascagoula-H orn 






4.137501 

-51+52 

Club 

47 03 13. 6 

-1.5 

12.1 

0.1 

12.0 

0.135496 

-35+36 

Pascagoula 

37 53 04.6 

-1.9 

02.7 

0.2 

02.5 

9. 788214 

-38+40 

Horn 

95 03 47.2 

-1.6 

45.6 

0.1 

45.5 

9. 998302 




- 5.0 


0.4 




Club-Horn 






4. 061211 


Club-Pascagoula 






4.271299 


Belle-Club 






3.993578 

-53+54 

Deer 

41 02 10. 7 

-0.9 

09.8 

0.1 

09.7 

0.182743 

-45+47 

Belle 

102 35 18. 5 

+ 4.8 

23.3 

0.0 

23.3 

9.989430 

-49+50 

Club 

36 22 26. 8 

+ 0.3 

27.1 

0.1 

27.0 

9. 773096 




+ 4.2 


0.2 




Deer-Club 






4.165751 


Deer-Belle 






3.949417 


Deer-Belle 






3.949417 

-60+61 

Ship 

33 10 60.2 

-3.6 

56.6 

0.1 

56.5 

0.261770 

-53+55 

Deer 

97 49 37. 9 

-2.8 

35.1 

0.1 

35.0 

9. 995936 

-46+47 

Belle 

48 59 22.9 

+ 5.7 

28.6 

0.1 

28.5 

9. 877722 




-0.7 


0.3 




Ship-Belle 






4.207123 


Ship-Deer 






4.088909 


Deer-Club 






4.165751 

-60+62 

Ship 

70 52 35.0 

-4.2 

30.8 

0.2 

30.6 

0.024657 

-54+55 

Deer 

56 47 27.2 

-1.9 

25.3 

0.1 

25.2 

9. 922555 

-48+49 

Club 

52 20 03. 5 

-f- 0. 8 

04.3 

0.1 

04.2 

9. 898501 




-5.3 


0.4 




Ship-Club 






4.112963 


Ship-Deer 


. 




4.088909 


Belle-Club 






3.993578 

-61+62 

Ship 

37 41 34. 8 

- 0.6 

34.2 

0.1 

34.1 

0.213655 

-45+46 

Belle 

53 35 55. 6 

-0.9 

54.7 

0.1 

54.6 

9. 905730 

-48+50 

Club 

88 42 30. 3 

+ 1.1 

31.4 

0.1 

31.3 

9.999890 




- 0.4 


0.3 




Ship-Club 






4.112963 


Ship-Belle 






4.207123 




































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


181 


Final computation of triangles —Continued 


Symbol 

Station 

Observed 

angle 

Cor¬ 

rec¬ 

tion 

Spher¬ 

ical 

angle 

Spher¬ 

ical 

excess 

Plane an¬ 
gle 

Loga¬ 

rithm 



o tit 

rt 

a 

it 

b t tt 



Deer-Ship 






4.088909 

- 63+64 

Biloxi Lighthouse 

48 11 17 . 4 

+ 1.3 

18.7 

0.1 

18.6 

0.127644 

- 55+57 

Deer 

96 30 31.2 

+ 6.3 

37 . 5 

0.1 

37.4 

9.997190 

- 59+60 

Ship 

35 18 04 . 4 

- 0.3 

04.1 

0.1 

04.0 

9 . 761833 




+ 7.3 


0.3 




Biloxi Lighthouse-Ship 






4.213743 


Biloxi Lighthouse-Deer 






3.978386 


Ship Island Lighthouse-Biloxi 






4 . 323999 


Lighthouse 







- 56+57 

Deer 

72 34 26 . 4 

- 3.7 

22.7 

0.1 

22.6 

0.020407 

— 66+67 

Ship Island Lighthouse 

25 29 62.1 

- 2.9 

59.2 

0.2 

59.0 

9.633980 

- 63+65 

Biloxi Lighthouse 

81 55 36.0 

+ 2.6 

38.6 

0.2 

38.4 

9.995675 




- 4.0 


0.5 




Deer-Biloxi Lighthouse 






3.978386 


Deer-Ship Island Lighthouse 






4 . 340081 


Ship Island Lighthouse-Biloxi 






4 . 323999 


Lighthouse 







- 58+59 

Ship 

95 44 07.0 

+ 2.2 

09.2 

0.1 

09.1 

0.002180 

- 66+68 

Ship Island Lighthouse 

50 31 41.2 

- 9.8 

31.4 

0.2 

31.2 

9.887564 

- 64+65 

Biloxi Lighthouse 

33 44 18.6 

+ 1.3 

19.9 

0.2 

19.7 

9 . 744612 




- 6.3 


0.5 




Ship-Biloxi Lighthouse 






4 . 213743 


Ship-Ship Island Lighthouse 






4.070791 


Deer-Ship 






4.088909 

- 67+68 

Ship Island Lighthouse 

25 01 39.1 

- 6.9 

32.2 

0.1 

32.1 

0.373636 

— 55+56 

Deer 

23 56 04 . 8 

+ 10.0 

14.8 

0.1 

14.7 

9 . 608246 

- 58+60 

Ship 

131 02 11.4 

+ 1.9 

13.3 

0.1 

13.2 

9.877536 




+ 5.0 


0.3 




Ship Island Lighthouse-Ship 






4.070791 


Ship Island Lighthouse-Deer 






4.340081 


t 


< 





















182 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


STATION CEDAR 


Final position computation, 








O 

9 

99 

a 

East base to west base 



84 

14 

41.9 

Second angle 

West base and Cedar 



+103 

55 

34.3 

a 

East base to Cedar 



188 

10 

16.2 

Ja 






+ 


29.4 







180 

00 

00.00 

a' 

Cedar to east base 



8 

10 

45.6 





First angle of triangle 

37 

26 

33.6 


O 

9 

n 







30 

14 

56.379 

Dauphin Island east 

X 

88 

08 

14.288 




base 





A<t> 

+ 

5 

51.937 


AX 

— 


58.262 

4>' 

30 

20 

48.316 

Cedar 

X' 

88 

07 

16. 026 



+1 









1st term 
2d and 3d 
terms 


— A<J> 


30 17 52 

99 

-351.9435 
+ 0.0063 


-351.9372 


s 

4.039349 

S 2 

8.0787 



COS a 

9.995568 

sin 2 a 

8.3054 

h 2 

5.093 

B 

8.511556 

C 

1.1712 

D 

2.332 

h 

2.546473 


7.5553 


7.425 




+0.0036 


+0.0027 


s 

sin a 

A' 

sec 

4.039349 
9.152688 
8. 509351 
0.063997 

AX 

sin §(^+$') 

1. 765385 
9. 702857 


1. 765385 


1. 468242 


99 


99 

AX 

-58.2620 

—Aa 

-29.39 


STATION CAT 








O 

9 

99 

a 

East base to west base 



84 

14 

41.9 

Second angle 

West base and Cat 



+ 53 

37 

23.0 

a 

East base to Cat 



137 

52 

04.9 

Aa 






— 

2 

08.4 







180 

00 

00.00 

a ' 

Cat to east base 




317 

49 

56.5 





First angle of triangle 

66 

01 

26.2 


O 

9 

99 






<t> 

30 

14 

56.379 

Dauphin Island east 
base 

X 

88 

08 

14.288 

A<{> 

+ 

4 

04.168 

AX 

+ 

6 

14.637 


30 

19 

00. 547 

Cat 

X' 

88 

14 

28.925 


§(<£+<£') 


1st term 
2d, 3d, and 
4th terms 


— 


O 9 99 

s 

4.006086 

s 2 

8.0122 



—h 

2.388 

30 16 58 

COS a 

9.870171 

sin 2 a. 

9. 6532 

h 2 

4. 776 

s 2 sin 2 a 

7.665 

99 

B 

8. 511556 

C 

1.1712 

D 

2.332 

E 

5. 917 

-244.2379 

h 

2.387813 


8.8366 


7.108 


5. 970 

+ 0.0700 




+0.0686 


+0.0013 


+0.0001 

-244.1679 










s 

sin a 
A' 

sec </>' 


AX 


4.006086 
9.826619 
8.509352 
0.063865 

AX 

sin K<£+^') 

2. 405922 


9 9 


+254.6373 

—Aa 


2.405922 
9. 702663 


2.108585 


+128.40 


































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 183 


secondary triangulation 


STATION CEDAR 












O 

/ 

tf 

a 

West base to east base 





264 

11 

22.1 

Third angle 

Cedar and east base 






- 38 

37 

52.4 

a 

West base to Cedar 






225 

33 

29.7 

Ja 










+ 

3 

49.5 











180 

00 

00.00 

a' 

Cedar to west base 






45 

37 

19.2 


O 


t 


n 









. 30 


14 

21. 492 

Dauphin Island west 

X 

88 

14 

51.034 

J<J> 

+ 


6 

26. 825 




- 

7 

35.008 


30 


20 

48.317 


Cedar 

y 

88 

07 

16.026 


O / 

tt 

s 

4.230995 

s 2 

8.4620 



-h 

2.588 


30 17 

35 

COS a 

9. 845212 

sin 2 a 

9. 7073 

h 2 

5.176 

s 2 sin 2 a 

8.169 


n 


B 

8.511557 

C 

1.1711 

D 

2.331 

E 

5.917 

1st term 

-387.0473 

h 

2.587764 


9.3404 


7.507 


6.674 

2d, 3d, andl 
4 th terms / 

+ 0.2227 






+0.2190 


+0.0032 


+0.0005 

— d<f> 

-386.8246 









% 



s 

sin a 
A' 

sec 4 V 


J\ 


4. 230995 
9. 853676 
8. 509351 
0.063997 

JX 

sin 4(^+0') 

2.658019 


// 


-455.0080 

—da 


2.658019 
9. 702795 


2.360814 


-229.52 


STATION CAT 








0 

/ 

ft 

a 

West base to east base 



264 

11 

22.1 

Third angle 

Cat and east base 



- 60 

21 

11.1 

a 

West base to Cat 



203 

50 

11.0 

Ja 






+ 

1 

11.7 







180 

00 

00.00 

a' 

Cat to west base 



23 

51 

22.7 


O 

/ 

ft 






<f> 

30 

14 

21. 492 

Dauphin Island west 
base 

X 

88 

14 

51. 034 

J<f> 

+ 

4 

39. 055 

A\ 

- 

2 

22.109 

V 

30 

19 

00.547 

Cat 

V 

88 

12 

28.925 


h(<J>+4>') 


1st term 
2d.3d,and' 
4th terms 

— J<f> 


30 16 41 

n 

-279.0776 
+ 0.0231 


-279.0545 
s 

sin a 
A' 

sec <j>' 


s 

cos a 
B 


3.972888 
9.961280 
8.511557 


2.445725 


sin 2 a 
C 


7.9458 
9. 2130 
1.1711 


JX 


3. 972888 
9. 606517 
8. 509352 
0. 063865 

J\ 

sin b(<j>+<f >') 

2.152622 


// 


-142.1092 

—Ja 


8. 3299 
+0. 0214 


2.152622 
9. 702600 


h 2 

D 


4.891 

2.331 


7. 222 
+0.0017 


-h 

s 2 sin 2 a. 
E 


1.855222 

n 

-71.65 


2. 445 
7.159 
5.917 


5.521 









































































184 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final position computation, 


STATION PINS 









/ 

tt 

a 

Cedar to west base 



45 

37 

19.2 

Second angle 

West base and Pins 



+ 53 

39 

37.8 

a 

Cedar to Pins 




99 

16 

57.0 

Ja 






— 

5 

43.8 







180 

00 

00.00 

a' 

Pins to Cedar 




279 

11 

13.2 





First angle of triangle 

58 

45 

27.5 


O 

t 

tt 




• 


<i> 

30 

20 

48.317 

Cedar 

X 

88 

07 

16.026 

J<£ 

+ 

1 

35.914 


JX 

+ 

11 

20. 225 

<f>’ 

30 

22 

24. 231 

Pins 

X' 

88 

18 

36.251 









+1 



O / tt 

30 21 36 

n 

s 

COS a 

B 

4.264911 
9.207641 
8.511550 

S 2 

sin 2 a 

C 

8.5298 
9.9885 
1.1729 

h 2 

D 

3.968 
2.332 

-h 

s 2 sin 2 a 
E 

1.984 

8.518 

5. 919 

1st term 

2d,3d, and \ 
4th terms / 

-96.4055 

+ 0.4916 

h 

1.984102 


9.6912 

+0.4911 


6.300 

+0.0002 


6.421 

+0.0003 

— A<f> * 

-95.9139 










s 

sin a 

A' 

sec <f)' 

4.264911 

9.9942754 
8.5093503 
0.064116 

JX 

sin $(<£+<£') 

2.832653 
9. 703663 


2.8326527 


2. 536316 


tt 


tt 

JX 

+680.225 

—Aa 

+343.81 


station grand 








© 

t 

tt 

a 

Pins to west base 



337 

56 

40.7 

Second angle 

West base and Grand 



+ 85 

13 

04.4 

a 

Pins to Grand 



* 

63 

09 

45.1 

Aa 






— 

3 

32.3 







180 

00 

00.00 

a' 

Grand to Pins 




243 

06 

12.8 





First angle of triangle 

54 

52 

03.0 


O 

t 

tt 







30 

22 

24.231 

Pins 

X 

88 

18 

36.252 

A<J> 

— 

3 

04.656 


JX 

+ 

7 

00.241 

V 

30 

19 

19.575 

Grand 

X' 

88 

25 

36.493 



o ttr 

S 

4.099752 

s 2 

8.1995 



-h 

2.266 


30 20 52 

COS a 

9.654620 

sin 2 a 

9.9010 

h 2 

4.532 

s 2 sin 2 a 

8.100 

ft 

B 

8.511548 

C 

1.1734 

D 

2.333 

E 

5.919 

1st term 

+ 184. 4676 

h 

2.265920 


9.2739 


6.865 


6.285 

2d,3d, and 1 
4th terms / 

+ 0.1884 




+0.1879 


+0.0007 


-0.0002 

— A<j> 

+184.6560 










s 

sin a 

A' 

sec «£' 

4.099752 

9.9505064 
8.509352 

0.063888 

JA 

sin %(</>+<!>') 

2.623498 
9. 703504 


2.6234984 


2.327002 


tt 


tt 

JX 

+ 420. 2409 

—Aa 

+212.32 







































































APPLICATION OP LEAST SQUARES TO TRIANGULATION. 185 


secondary triangulation —Continued 

STATION PINS 


Third angle 


a 

Act 


<f> 

A<f> 

<t>' 


§(<£+<£') 


1st term 
2d, 3d,and 
4th terms . 

— A<f> 


West base to Cedar 
Pins and Cedar 

West base to Pins 


Pins to west base 


30 


+ 


14 

8 


30 

O / 

30 18 23 

tt 

-482. 7974 
+ 0.0587 


22 


21. 492 
02. 739 


24.231 


Dauphin Island west 
base 


Pins 


X 

A\ 

X' 


o 

t 

tt 

225 

33 

29.7 

- 67 

34 

55.3 

157 

58 

34.4 

— 

1 

53.7 

180 

00 

00 . 00 

337 

56 

40.7 

88 

14 

51.034 

+ 

3 

45 . 218 


88 


18 


36. 252 


-482.7387 
s 

sin a 
A' 
sec 


s 

4.205115 

«2 

8. 4102 



-h 

2.684 

COS a 

9.967093 

sin 2 a 

8.1480 

h 2 

5.368 

s 2 sin 2 a 

7.558 

B 

8.511557 

C 

1.1711 

D 

2.331 

E 

5.917 

h 

2.683765 


8. 7293 


7.699 


6.159 




+0.0.536 


+0.00.50 


+0.0001 


A\ 


4.205115 

9.5740212 
8.5093503 
0.064116 

A\ 

sin *(<£+<£') 

2.3526025 


tf 


+225. 2177 

—Aa 


2.352602 
9. 702967 


2.0,55569 


+ 113.65 


STATION GRAND 


Third angle 

CL 

Acl 


4 > 

A<f> 

4 >' 

$((£+<£') 


1st term 
2d, 3d, andl 
4th terms / 

-A<f> 


West base to Pins 
Grand and Pins 

West base to Grand 


Grand to west base 

O 

30 


+ 


14 

4 


30 

O t 

30 16 50 

tr 

-298.5266 
+ 0.4435 


19 


21.492 
58.083 


19.575 


Dauphin Island west 
base 


Grand 


X 

A\ 

X' 


O 

157 
- 39 

/ 

58 

54 

ft 

34.4 

53.1 

118 

03 

41.3 

— 

5 

25.5 

180 

00 

00.00 

297 

58 

15.8 

88 

14 

51.034 

+ 

10 

45. 459 


88 


25 


36 . 493 


-298.0831 
S 

sin a 
A' 
sec 


S 

4.290942 

s 2 ' 

8.5818 



-h 

2.475 

COS a 

9.672484 

sin 2 at 

9. 8914 

h 2 

4.950 

s 2 sin 2 a 

8.473 

B 

8.511557 

C 

1.1711 

D 

2.331 

E 

5.917 

h 

2.474983 


9.6443 


7.281 


6.865 




+0. 4409 


+0.0019 


+0.0007 


A\ 


4.290942 

9.945687 

8.509352 

0.063888 

A\ 

sin §(<£+<£') 

2.809869 


tf 


+645. 4595 

—Act 


2.809869 
9. 702633 


2.512502 

tf 

+325. 46 







































































186 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final 'position computation , 


STATION PETIT 


o 

297 
+ 59 

9 

58 

11 

n 

15.8 

52.0 

357 

+ 

10 

07.8 

11.6 

180 

00 

00.00 

177 

10 

19.4 

81 

41 

30.1 

88 

25 

36.493 
23.006 

88 

25 

13.487 


Second angle 


a 

Aa 


<t> 

A<f> 

F 


Grand to west base 
West base and Petit 

Grand to Petit 


Petit to Grand 


First angle of triangle 


30 


30 


/ 

19 

6 


12 


19.575 
44.068 


35.507 


Grand 

Petit 


X 

A\ 


sin a 
A' 

sec F 


A\ 


4.095429 
8.693666 
8.509354 
0.063392 

A\ 

sin i(^+$') 

1.361841 


99 


-23.006 

—Aa 


1.361841 
9.702445 


1.064286 


-11.59 



O t 99 

S 

4.095429 

S 2 

8.1909 




30 15 58 

cos a 

9.999470 

sin 2 a 

7.3873 

h 2 

5.213 

99 

B 

8.511551 

C 

1.1725 

D 

2.332 

1st term 

+404.0639 

h 

2.606450 


6. 7507 


7.545 

2d and 3d 1 
terms / 

+ 0.0041 




+0.0006 


+0.0035 

-A<1> 

+404.0680 








STATION HORN 


O 

357 
+ 61 

9 

10 

04 

99 

07.8 

37.0 

58 

14 

44.8 

— 

5 

18.1 

180 

00 

00.00 

238 

09 

26.7 

38 

16 

35.3 

88 

25 

36. 493 

+ 

10 

30.972 


Second angle 


a 

Jot 



O 

9 

99 


30 

19 

19.575 

A(j> 

— 

5 

39. 559 

<V 

30 

13 

40.016 


1st term 
2d, 3d, and 1 
4 th terms / 

— A<f> 


Grand to Petit 
Petit and Horn 

Grand to Horn 


Horn to Grand 


First angle of triangle 
Grand 


Horn 


X 

A\ 


88 


36 


07. 465 


30 16 30 

• n 

+339.1338 
+ 0.4253 


+339.5591 


s 

sin a 
A' 
sec <t> 


s 

4.297606 

$ 2 

8.5953 



-h 

2.530 

cos a 

9. 721214 

sin 2 a 

9.8592 

h 2 

5.061 

s 2 sin 2 a 

8. 454 

B 

8. 5115514 

C 

1.1725 

D 

2.332 

E 

5.918 

h 

2.5303714 


9.6270 


7.393 


6.902 




+0. 4236 


+0.0025 


-0.0008 


A\ 


4.297606 
9.929579 
8.509354 
0.063471 

A\ 

sin 

2. 800010 


99 


+630.9719 

—Aa 


2.800010 
9. 702560 


2. 502570 

n 

+318.10 





































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 187 


secondary triangulation —Continued 

STATION PETIT 








O 

/ 

n 

a 

West base to Grand 



118 

03 

41.3 

Third angle 

Petit aftd Grand 



- 39 

06 

38.4 

a 

West base to Petit 



78 

57 

02.9 

Ja 






— 

5 

13.3 







180 

00 

00.00 

a' 

Petit to west base 



258 

51 

49.6 









-.1 


o 

9 

ft 







30 

14 

21. 492 

Dauphin Island west 

X 

88 

14 

51.034 





base 





J<f> 

— 

1 

45.985 


J\ 

+ 

10 

22.452 


30 

12 

35.507 

Petit 

X' 

88 

25 

13. 486 









+1 



0 9 99 

s 

4.229486 

52 

8.4590 



-h 

2.024 


30 13 28 

cos a 

9.282513 

sin 2 a 

9.9837 

h 2 

4.047 

s 2 sin 2 a 

8. 443 


99 

B 

8.511557 

C 

1.1711 

D 

2.331 

E 

5.917 

1st term 

+ 105. 5737 

h 

2.023556 


9.6138 


6. 378 


6.384 

2d, 3d, and 1 
4 th terms / 

+ 0.4109 

. 



+0. 4109 


+0. 0002 


-0.0002 

— J<f> 

+ 105.9846 










s 

sin a 

A' 

sec <{>' 

4.229486 
9.991874 
8.509354 
0.063392 

JX 

sin $(<£+<£') 

2.794106 
9. 701905 


2. 794106 


2.496011 


99 


99 

J\ 

+622. 4522 

—Ja 

+313.34 


STATION HORN 







O 

9 

99 

CL 

Petit to Grand 



177 

10 

19.4 

Third angle 

Horn and Grand 



- 80 

38 

48.3 

a 

Petit to Horn 



96 

31 

31.1 

Ja 





— 

5 

29.1 






180 

00 

00.00 

a ' 

Horn to Petit 



276 

26 

02.0 


O 

9 

99 







30 

12 

35.507 

Petit 

X 

88 

25 

13. 487 

J<f> 

+ 

1 

04.509 


J\ 

+ 

10 

53.978 


30 

13 

40.016 

% 

Horn 

X' 

88 

36 

07.465 



0 9 99 

s 

4.245560 

s 2 

8.4912 



-h 

1.813 

*(*+*') 

30 13 08 

cos a 

9.055539 

sin 2 a 

9.9943 

h 2 

3.625 

s 2 sin 2 a 

8. 485 

99 

B 

8.511559 

C 

1.1706 

D 

2. 331 

E 

5.916 

1st term 

-64.9618 

h 

1.812658 


9.6561 


5.956 


6.214 

2d, 3d, and \ 
4 th terms / 

+ 0.4533 




+0.4530 


+0.0001 


+0.0002 

— J<f> 

-64.5085 

# 









S 

sin a 

A' 

sec <£' 

+4 
4.245560 
9.9971774 
8.5093541 
0.063471 

JX 

sin £(<£+<£') 

2.815563 
9.701830 


2.815563 


2.517393 


99 


99 

ji 

+653.9778 

—Ja 

+329.15 










































































188 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation , 

STATION PASCAGOULA 


Second angle 

a 

Joe 


<t> 

jrfj 


1st term 
2d, 3d, and \ 
4th terms / 


— J<f> 


Grand to Horn 
Horn and Pascagoula 

Grand to Pascagoula 


Pascagoula to Grand 


30 


+ 


19 

1 


30 


30 20 00 

It 

-81. 2232 
+ 0.2256 


20 


19.575 
20.998 


40. 573 


First angle of triangle 
Grand 
Pascagoula 


X 

JX 


0 

58 
+ 43 

I 

14 

14 

n 

44.8 

20.8 

101 

29 

05.6 

— 

3 

52.8 

180 

00 

00. 00 

281 

25 

12.8 

97 

55 

59.1 

88 

25 

36. 493 

+ 

7 

40. 886 


88 


33 


17.379 


-80.9976 
s 

sin a 
X' 

sec ft 


s 

4.099038 

S 2 

8.1981 



— h 

1.910 

COS a 

9.299092 

sin 2 a 

9.9824 

h 2 

3.820 

s 2 sin 2 a 

8.180 

B 

8.511551 

C 

1.1725 

•D 

2.332 

E 

5.918 

h 

1.909681 


9. 3530 


6.152 


6.008 




+0.2254 


+0.0001 


+0.0001 


JX 


4.099038 
9.991216 
8.509351 

0.063988 

JX 

sin £(<£+<£') 

2. 663593 


It 


+460.8859 

— Ja 


2.663593 
9. 703317 


2.366910 

It 

+232. 76 


STATION BELLE 








0 

1 

It 

a 

Pascagoula to Horn 



19 

21 

11.9 

Second angle 

Horn and Belle 




+ 69 

43 

26.3 

a 

Pascagoula to Belle 



89 

04 

38.2 

Ja 






— 

5 

01.7 







180 

00 

00.00 

a ' 

Belle to Pascagoula 



268 

59 

36.5 





First angle of triangle 

48 

58 

48.4 


O 

/ 

It 







30 

20 

40. 573 

Pascagoula 

X 

88 

33 

17. 379 

J<f> 

— 


08. 724 


JX 

+ 

9 

57.281 

F 

30 

20 

31. 849 

Belle 

X' 

88 

43 

14. 660 









+2 


§(<£+<£') 


1st term 
2d and 3d 
terms 


— J4> 


Oft! 

s 

4.202907 

$ 2 

8.4059 



30 20 36 

COS a 

8.206975 

sin 2 a 

9. 9999 

h 2 

1.844 

It 

B 

8.511550 

C 

1.1729 

D 

2.332 

+8.3451 

h 

0.921432 


9. 5787 


4.176 

+0. 3790 




+0.3790 



‘+8.7241 








s 

sin a 
A' 

sec <£' 


JX 


4.202907 
9.999944 
8.509351 
0.063977 

JX 

sin $(<£+<£') 

2. 776179 


It 


+597. 2814 

—Ja 


2. 776179 
9. 703447 


2. 479626 


+301. 73 




































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 189 


secondary triangulation —Continued 

STATION PASCAGOULA 


Third angle 


a 

Joe 


4> 


<y 

$(<£+<£') 


1st term 
2d,3d,andl 
4th terms / 


— J4> 


Horn to Grand 
Pascagoula and Grand 

Horn to Pascagoula 


Pascagoula to Horn 


30 


+ 


13 

7 


40.016 
00.557 


30 


30 17 10 

99 

-420. 5920 
+ 0.0345 


20 


40.573 


Horn 

Pascagoula 


X 

JX 


0 

238 
- 38 

/ 

09 

49 

99 

26.7 

40.5 

199 

19 

46.2 

+ 

1 

25.8 

180 

00 

00.00 

19 

21 

12.0 

-.1 

88 

36 

07. 465 

— 

2 

50.086 


88 


33 


17.379 


-420.5575 


S 

4.137501 

S 2 

8.2750 



— h 

2.624 

COS a 

9.974802 

sin 2 a 

9.0396 

h 2 

5.248 

s 2 sin 2 a 

7. 315 

B 

8.511558 

C 

1.1709 

D 

2.331 

E 

5.917 

h 

2.623861 


8. 4855 


7.579 


5.856 




+0.0306 


+0.0038 


+0.0001 


s 

sin a 

A' 

sec 4 >' 

4.137501 

9. 519828 
8.509351 
0.063988 

JX 

sin $(<£+<£') 

2.230667 
9. 702706 


2.230668 


1.933373 


99 


n 

JX 

-170.0858 

—Ja 

-85.78 


STATION BELLE 







o 

r 

99 

a 

Horn to Pascagoula 


199 

19 

46.2 

Third angle 

Belle and Pascagoula 


- 61 

17 

-45.8 

a 

Horn to Belle 



138 

02 

00.4 

Ja 





— 

3 

35.4 






180 

00 

00.00 

a' 

Belle to Horn 



317 

58 

25.0 








-.1 


O 

9 

99 






<t> 

30 

13 

40.016 

Horn 

X 

88 

36 

07. 465 

J<f> 

+ 

6 

51.834 


JX 

+ 

7 

07.197 


30 

20 

31.850 

Belle 

X' 

88 

43 

14.662 




-1 







h(4>+4>’) 

0 9 99 

30 17 06 

99 

S 

COS a 

B 

4.232071 
9.871302 
8.511558 

s 2 

sin 2 a 

C 

8.4642 
9.6505 
1.1709 

h 2 

D 

5.230 
2.331 

-h 

s 2 sin 2 w 
E 

2.615 
8.115 

5.917 

1st term 

-412.0321 

h 

2.614931 


9.2856 


7.561 


6. 647 

2d,3d, andl 
4th terms / 

+ 0.1976 




+0.1936 


+0.0036 


+0.0004 

-J4> 

-411.8345 










S 

sin a 

A' 

sec <f>' 

4.232071 

9. 825229 
8. 509351 
0.063977 

JX 

sin 

2.630628 
9. 702690 


2. 630628 


2.333318 


• 99 


99 

JX 

+427.1968 

—Ja 

+215.44 










































































190 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Final position computation , 


STATION CLUB 


Second angle 


a 

Joe 


<f> 

J<ft 

<y 




1st term 
2d and 3d \ 
terms / 
— J<j> 


Belle to Horn 
Horn and Club 

Belle to Club 


Club to Belle 


First angle of triangle 


30 


30 


20 

5 


15 


31.849 
19.873 


11.976 


Belle 

Club 


X 

JX 


s 

sin a 
A' 

sec <f>' 


JX 


3.993578 
8.424985 
8.509353 
0.063584 

J\ 

sin 

0.991500 


;/ 


-9.8062 

—Jot 


0.991500 
9. 702856 


0.694356 


-4.95 


o t tr 

s 

3.993578 

S 2 

7.9872 



30 17 52 

COS a 

9.999846 

sin 2 a 

6.8500 

h 2 

5.010 

tr 

B 

8.511550 

C 

1.1729 

D 

2.332 

+319.8704 

h 

2.504974 


6.0101 


7.342 

+ 0.0023 




+0.0001 


+0.0022 

+319.8727 








O 

/ 

tt 

317 

58 

24.9 

+40 

30 

06.4 

358 

28 

31.3 

+ 


05.0 

180 

00 

00.00 

178 

28 

36.3 

105 

43 

54.1 

88 

43 

14.662 

— 


09.806 

88 

43 

04.856 


STATION DEER 


O 

t 

tr 

358 

28 

31.3 

+102 

35 

23.3 

101 

03 

54.6 

— 

2 

45.3 

180 

00 

00.00 

281 

01 

09.3 

41 

02 

09.8 

88 

43 

14.662 

+ 

5 

27.087 

88 

48 

41. 749 


Second angle 


a 

Joe 


<t> 

j<t> 


<y 


1st term 
2d and 3d 


terms } 
-A4> 


Belle to Club 
Club and Deer 

Belle to Deer 


Deer to Belle 


30 


+ 


30 


20 


21 


31. 849 
55.361 


27.210 


First angle of triangle 
Belle 


Deer 


X 

jx 

X' 


s 

sin a 
A' 

sec <}>' 


J\ 


3.949417 
9.991850 
8.509351 
0.064045 

J\ 

sin §(<£+<£') 

2.514663 


tr 


+327.0872 

—Ja 


2.514663 
9. 703531 


2.218194 


+165. 27 


o r tr 

s 

3.949417 

« 2 

7.8989 



30 21 00 

COS a 

9.283133 

sin 2 a 

9.9837 

h 2 

3. 488 

tr 

B 

8.511550 

C 

1.1729 

D 

2.332 

-55.4752 

h 

1. 744100 


9.0555 


5.820 

+ 0.1137 




+0.1136 


+0.0001 

-55.3615 








































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 191 


secondary triangulation —Continued 

STATION CLUB 


Horn to Belle 
Club and Bello 

Horn to Club 


Club to Horn 


30 


+ 


30 


13 

1 


15 


40.016 
31. 961 


11.977 

-1 


S 

sin a 
A' 

sec <t>' 


JX 


Horn 

Club 



O / // 

s 

4.061211 

S 2 

b(<f>+<f>') 

30 14 26 

COS a. 

9.391708 

sin 2 « 

ft 

B 

8.511558 

C 

1st term 

2d,3d,and \ 
4 th terms / 
— J<f> 

-92.1459 

+ 0.1848 

h 

1.964477 


-91.9611 





8.1225 
9.9728 
1.1709 


4.061211 
9.986395 
8.509353 
0.063584 

JX 

sin £(<£+<£') 

2.620543 


tt 


+417.3909 

—Joe 


9. 2662 
+0.1845 


2.620543 
9. 702113 


2.322656 

tt 

+210.21 



O 

138 

-33 

! 

02 

45 

tt 

00.4 

59.8 


104 

16 

00.6 


— 

3 

30.2 


180 

00 

00.00 


284 

12 

30.4 

X 

88 

36 

07.465 

JX 

+ 

6 

57. 391 

X' 

88 

43 

04.856 



-h 

1.964 

h 2 

3.929 

s 2 sin 2 a 

8.095 

D 

2.331 

E 

5.917 


6.260 


5.976 


+0.0002 


+0.0001 


STATION DEER 















o 

t 

tt 

a 

Club to Belle 









178 

28 

36.3 

Third angle 

Deer and Bello 









-36 

22 

27.1 

a 

Club to Deer 









142 

06 

09.2 

Joe 













— 

2 

50.0 














180 

00 

00.00 

a ' 

Deer to Club 









322 

03 

19.2 
















-.1 


O 



/ 


tt 









<f> 

30 



15 

11.976 


Club 

X 


88 

43 

04.856 

J<f> 

+ 



6 

15. 234 





JX 


+ 

5 

36.893 

<t>' 

30 



21 

27.210 


Deer 

X' 


88 

48 

41.749 


O / 

rr 

6 


4 

. 165751 

s 2 


8.3315 




—h 

2.574 

l(ef>+<y) 

30 18 

20 

COS a 

9.897138 

sin 2 a 

9.5767 

h 2 


5.149 

s 2 sin 2 a 

7.908 

tt 



B 

8.511556 

C 


1.1713 

D 


2.331 

E 

5. 918 

1st term 

-375.3574 

I 

L 

2.574445 



9.0795 



7.480 


6.400 

2d,3d, andl 
4th terms / 

+ 0.1233 








+0.1200 


+0.0030 


+0.0003 
















-Jef> 

-375.2341 















s 



4.165751 







* 




sin cx 



9.788345 











A' 



8.509351 


JX 


2.527492 






sec ef>' 



0.064045 

sin h(,<f>+ef>') 

9. 702957 









2. 527492 




2.230449 









rr 






tt 






JX 


+336.8930 


—Ja 

+170.00 






91865°—15-13 












































































192 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation , 


STATION SHIP 








O 

9 

99 

CL 

Deer to Club 




322 

03 

19.1 

Second angle 

Club and Ship 




+ 56 

47 

25.3 

CL 

Deer to Ship 




18 

50 

44.4 

Ja 






— 

1 

14.8 







180 

00 

00.00 

a' 

Ship to Deer 




198 

49 

29.6 





First angle of triangle 

70 

52 

30.8 


* o 

9 

99 






<t> 

30 

21 

27. 210 

Deer 

X 

88 

48 

41. 749 

J<f> 

— 

6 

17.189 


J\ 

+ 

2 

28. 277 


30 

15 

10.021 

Ship 

X' 

88 

51 

10. 026 



0 9 99 

s 

4.088909 

s 2 

8.1778 




30 18 19 

COS a 

9.976071 

sin 2 a 

9.0184 

h 2 

5.153 

99 

B 

8.511549 

C 

1.1731 

D 

2. 332 

1st term 

+377.1629 

h 

2.576529 


8.3693 


7.485 

2d and 3d \ 
terms / 

+ 0.0265 




+0.0234 


+0.0031 


+377.1894 








s 

sin a 

A' 

sec 

4.088909 
9. 509230 
8.509353 
0.063581 

J\ 

sin 

2.171073 
9. 702952 


2.171073 


1.874025 


99 


99 

J\ 

+148.2767 

—Ja 

+74. 82 


STATION BILOXI LIGHTHOUSE 








o 

9 

99 

CL 

Deer to Ship 




18 

50 

44.4 

Second angle 

Ship and Biloxi Lighthouse 


+ 96 

30 

37.5 

a 

Deer to Biloxi Lighthouse 


115 

21 

21.9 

Ja 






— 

2 

42.9 







180 

00 

00.00 

a' 

Biloxi Lighthouse to Deer 


295 

18 

39.0 





First angle of triangle 

48 

11 

18.7 


O 

/ 

99 







30 

21 

27. 210 

Deer 

X 

88 

48 

41. 749 

J<t> 

+ 

2 

12. 209 


J\ 

+ 

5 

22.071 

<y 

30 

23 

39. 419 

Biloxi Lighthouse 

X' 

88 

54 

03.820 




1st term 
2d, 3d, and 1 
4th terms / 

— J<j> 


0 9 99 

s 

3.978386 

S 2 

7.9568 



—h 

2.121 

30 22 33 

COS a 

9.631690 

sin 2 a 

9.9120 

h 2 

4.243 

s 2 sin 2 a 

7.869 

99 

B 

8.511549 

C 

1.1731 

D 

2.332 

E 

5.919 

-132. 3198 

h 

2.121625 


9. 0419 


6.575 


5.909 

+ 0.1107 




+0.1102 


+0.0004 


+0.0001 

-132.2091 










s 

sin a 

A' 

sec <f>' 

3.978386 
9.956007 
8.509350 
0.064209 

J\ 

sin i(<f>+<f>') 

2.507952 
9.703868 


2. 507952 


2. 211820 


99 


99 

J\ 

+322. 0713 

—Ja 

+162.86 


































































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 193 


secondary triangulation — Continued 

STATION SHIP 







O 

t 

tt 

a 

Club to Deer 



142 

06 

09.2 

Third angle 

Ship and Deer 



- 52 

20 

04.3 

a 

Club to Ship 



89 

46 

04.9 

Act 





— 

4 

04.4 






180 

00 

00.00 

a' 

Ship to Club 



269 

42 

00.5 

-.1 


O 

t 

tt 







30 

15 

11.976 

Club 

X 

88 

43 

04.856 

A<f> 

— 


01.955 


A\ 

+ 

8 

05.170 

V 

30 

15 

10.021 

Ship 

X' 

88 

51 

10.026 



O t tt 

5 

4.112963 

s 2 

8.2260 


30 15 11 

COS a. 

7. 607314 

sin 2 a 

0.0000 

tt 

B 

8.511556 

C 

1.1713 

1st term 

+ 1.7054 

h 

0.231833 


9. 3973 

2d term 

+0. 2496 




+0. 2496 

— A<f> 

+1.9550 






s 

sin a 
A' 

sec <f>' 


A\ 


4.112963 

9. 9999964 

8. 5093531 

A\ 

0.0635812 

sin l(<t>+4>') 

2. 6858937 


tt 

+485.1697 

—Aa 


2.685894 
9. 702275 


2. 388169 

n 

+ 244.44 


STATION BILOXI LIGHTHOUSE 


Third angle 

a 

Act 


A<f> 

<y 

1st term 
2d,3d,and \ 
4th terms / 

—A4> 


Ship to Deer ' 

Biloxi Lighthouse and Deer 

Ship to Biloxi Lighthouse 


Biloxi Lighthouse to Ship 


O 

30 


+ 


15 

8 


10.021 
29.398 


30 

Fixed value 
o t n 
30 19 25 

n 

-509.4353 
+ 0.0376 


23 


39. 419 

39. 419 


Ship 

Biloxi Lighthouse 



O 

198 
- 35 

/ 

49 

18 

tt 

29.6 

04.1 


163 

31 

25.5 


— 

1 

27.7 


180 

00 

00.00 


343 

29 

57.8 

-.1 

X 

88 

51 

10.026 

A\ 

+ 

2 

53.794 

X' 

88 

54 

03. 820 

03. 820 


-509.3977 
s 

sin 
A' 

sec $’ 


s 

4.213743 

s 2 

8.4275 



COS a 

9.981790 

sin 2 a 

8.9056 

h 2 

5.414 

B 

8.511556 

C 

1.1713 

D 

2.331 

h 

2. 707089 


8.5044 


7.745 




+0.0319 


+0.0056 


—h 

s 2 sin 2 a 
E 


2.707 

7.333 

5.918 


5.958 

+0.0001 


A\ 


4. 213743 
9.452733 
8. 509350 
0.064209 


2. 240035 

ft 

+ 173.7941 


A\ 

sin §(<£+<£') 


—Aa. 


2.240035 
9. 703190 


1.943225 

!9 

+87.74 

































































194 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Final position computation, 

STATION SHIP ISLAND LIGHTHOUSE 








O 

/ 

tt 

a 

Biloxi Lighthouse to Ship 


343 

29 

57.7 

Second angle 

Ship and Ship Island Lighthouse 


+ 33 

44 

19.9 

a 

Biloxi Lighthouse to Ship Island Lighthouse 


17 

14 

17.6 

Aa 






— 

1 

57.9 







180 

00 

00.00 

ex' 

Ship Island Lighthouse to Biloxi Lighthouse 


197 

12 

19.7 - 





First angle of triangle 

50 

31 

31.4 

<t> 

O 

30 

/ 

23 

ft 

39.419 

Biloxi Lighthouse 

X 

88 

54 

03. 820 

dcf> 

— 

10 

54.078 


A\ 

+ 

3 

53. 644 

<t>' 

30 

12 

45.341 

Ship Island Light- 

X' 

88 

57 

57. 464 





house 







O t tt 

30 18 12 

tt 

s 

COS a 

B 

4.323999 
9.980040 
8.511546 

S 2 

sin 2 a 

C 

8.6480 
8.9436 
1.1738 

h 2 

D 

5.631 

2.332 

-h 

$ 2 sin 2 a 
E 

2.816 
7.592 

5.919 

1st term 

2d, 3d, and "1 
4th terms / 

+654.0109 

+ 0.0673 

h 

2.815585 


8. 7654 

+0.0583 


7.963 

+0.0092 


6.327 

-0.0002 

— A<j> 

+654.0782 










5 

sin a 

A' 

sec <j>' 

4.323999 
9.471798 
8. 509354 
0.063404 

A\ 

sin K<f>+<f>') 

2.368555 
9. 702930 


2.368555 


2.071485 


tt 


tf 

A\ 

+233.6442 

— Aa 

+117.89 




































APPLICATION OF LEAST SQUARES TO TRIANGULATION 


195 


secondary triangulation —Continued 

STATION SHIP ISLAND LIGHTHOUSE 








O 

/ 

It 

CL 

Ship to Biloxi Lighthouse 



163 

31 

25.5 

Third angle 

Ship Island Lighthouse and Biloxi Lighthouse 


- 95 

44 

09.2 

CL 

Ship to Ship Island Lighthouse 


67 

47 

16.3 

Ja 



•v 




3 

25.2 







ISO 

00 

00.00 

a' 

Ship Island Lighthouse to Ship 


247 

43 

51.1 

4> 

O 

30 

/ 

15 

n 

10.021 

Ship 

X 

88 

51 

10.026 

A<f> 

— 

2 

24.681 


A\ 

+ 

6 

47.438 

V 

30 

12 

45.340 

Ship Island Light- 

X' 

88 

57 

57.464 




+1 

house 




57 . 464 


Fixed value 

45 . 341 








o t n 

30 13 58 

n 

S 

COS a 

B 

4.070791 
9. 577534 
8. 511556 

s 2 

sin 2 a 

C 

8.1416 
9.9330 
1.1713 

h 2 

D 

4.320 

2.331 

-h 

s 2 sin 2 a 
E 

2.160 
8.075 

5.918 

1st term 

2d,3d, and \ 
4th terms / 

+144.5043 

+ 0.1765 

h 

2.159881 


9. 2459 

+0.1762 


6.651 

+0.0004 


6.153 

-0.0001 

— Ad> 

+144.6808 










4.070791 

9.966513 
8.509354 

A\ 

2. 610062 

0.063404 

sin £($+<£') 

9. 702011 

2.610062 


2. 312073 

n 


n 

+ 407. 4385 

—Aa 

+205.15 




































196 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

ADJUSTMENTS BY THE ANGLE METHOD 

If the adjustment be made according to the angle method* the 
complications due to the presence of the z’s are avoided. An angle 
is the difference of two directions and the observation equation for 
an observed angle is the difference of the observation equations of its 
two sides, and in taking the difference the z drops out. To illustrate 
this suppose that at station Gunner, Figure 6, page 104, the following 
angles were observed: Duck to Indian Point, Indian Point to Larrabee, 
Larrabee to Mam, Mam to Lubec Channel Lighthouse, and Lubec 
Channel Lighthouse to Lubec Church spire. Call the corrections to 
the observed angles u v u 2 , u 3 , u 4 , and u 5 respectively, and suppose the 
observed and assumed values to be as given on page 115. Then 

U l =v 2 -v l =- 6131^0! + 1800<Wj - 2.2 
u 2 = v 3 — v 2 = — 1997^ + 236^ x + 7.0 
In a similar way, 

u 3 = -2732^+ 85^-1.2 

u 4 = - 5565^ - 2445^ + 3.5 
u 5 = + 5886^ + 5413(5^ +18.4 

These contain no z’s and the normal equations may be formed in the 
usual way. 

Observation equations of this kind would arise when at an unknown 
point angles are taken on known points, as for example when angles 
are taken with a sextant from a point off-shore to determine its posi¬ 
tion, and for such observations the angle method is both easier and 
more logical than the direction method. 


♦Wright and Hayford, Adjustment of observations, p. 180. 



APPLICATION OF LEAST SQUARES TO TRIANGULATION. 197 
ADJUSTMENT OF VERTICAL OBSERVATIONS 
GENERAL STATEMENT 

When reciprocal vertical observations are made over the lines of a 
triangulation scheme a computation of the differences of elevations is 
made by the usual Coast and 


Geodetic Survey fonnula. For 
an account of these observa¬ 
tions and of the method of 
computation, see United States 
Coast and Geodetic Survey 
Special Publication No. 19, 
page 140 et seq. As there are 
always several lines from each 
station, rigid conditions are 
present in the figure. Thus it 
becomes necessary to make an 
adjustment of the observed 
values by the method of least 
squares. In the following fig¬ 
ure the differences of eleva¬ 
tions as observed are first com¬ 
puted and then the results are 
adjusted by the method em¬ 
ployed in the United States 
Coast and Geodetic Survey. 

The formula used in the fol¬ 
lowing computations is the one 
given in Special Publication 
No. 19, mentioned above. On 
pages 205 et seq. there is given 
a new development of the for¬ 
mula that takes into account 
some of the small terms that 
are needed in computation over 
form of the new fonnula differs 
computation. 



longer and higher lines. The final 
slightly from the one used in this 




198 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, 


Computation of elevations from reciprocal observations 


Station 1 

Pollywog 

Pollywog 

Pack Sad- 

Pack Sad- 

Elk 

Elk 




die 

die 



Station 2 

Pack Sad- 

Elk 

High Hi- 

Elk 

Long Ridge 

High Hi- 


die 


vide 

- 


vide 


O f ft 

Off/ 

O t ft 

O f ff 

O f ff 

O f ff 

Ci 

90 04 13 

90 57 44 

90 30 00 

91 53 50 

88 33 16 

89 15 25 

C2 

90 04 59 

89 11 15 

89 37 33 

88 10 59 

91 36 54 

90 51 04 

C 2 -C 1 

+ 40 

- 1 40 29 

- 52 27 

- 3 42 51 

+ 3 03 38 

+ 1 35 39 

KC 2 -C 1 ) 

+ 23 

53 14 

- 26 14 

- 1 51 26 

+ 1 31 49 

+ 47 50 

tan KC 2 -C 1 ) 

G.04732 

8.18994 

7.88258 

8.51090 

8.42675 

8.1434S 

log s 

4.27444 

4.29253 

4.15543 

3.98144 

4.31524 

4.17150 

log s tan KC 2 -C 1 ) 

0.32176 

2.48247 

2.03801 

2.49234 

3.74199 

2.31498 

s tan KC 2 -C 1 ) 

+2.10 

-303. 72 

-109.15 

-310. 70 

+552.06 

+206.53 

Second term 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

Third term 

0.00 

0.03 

0. 01 

0.03 

0.06 

0.02 

ho—hi 

+2.10 

-303. 75 

-109.16 

-310. 73 

+552.12 

+200.55 

2 log s 

8.549 

8.585 

8.311 

7.963 

8.630 

8.343 

log p =9—2 logs 

0.451 

0.415 

0.689 

1.037 

0.370 

0.657 

p of h 2 —hi 

2.82 

2.60 

4.89 

10. 89 

2.34 

4.54 

Station 1 

High Hi- 

High Di- 

High Hi- 

Long Ridge 

Long Ridge 

Gordon 


vide 

vide 

vide 




Station 2 

Bald Hill 

Gordon 

Long Ridge 

Gordon 

Bald Hill 

Bald Hill 


O f ft 

Off/ 

O t ft 

O f ff 

0 rtr 

O t ft 

Ci 

89 29 43 

88 28 23 

88 12 34 

89 14 01 

91 27 22 

92 45 09 

C 2 

89 37 50 

91 40 22 

91 53 03 

90 52 54 

88 41 46 

87 21 10 

C 2 -C 1 

- 51 53 

+ 3 11 59 

+ 3 40 31 

+ 1 38 53 

- 2 45 36 

-5 23 59 

KC 2 -C 1 ) 

- 25 50 

+ 1 36 00 

+ 1 50 16 

+ 49 26 

- 1 22 48 

-2 42 00 

tan KC 2 -C 1 ) 

7.87759 

8.44611 

8.50632 

8.15777 

8.38184 

8.67357 

logs 

4.21540 

4.29195 

4.03443 

4.14252 

4.29013 

4.15348 

logs tanKC 2 -C 1 ) 

2.09305 

2. 73800 

2.54075 

2.30029 

2.67197 

2.82705 

s tan KC 2 -C 1 ) 

-123.89 

+547.09 

+347.34 

+199.66 

-469.86 

-671.51 

Second term 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

Third term 

0.01 

0.08 

0.05 

0.03 

0.06 

0.07 

h2—hi 

-123.90 

+547.17 

+347.39 

+199.69 

-469.92 

-671.58 

2 log s 

8.431 

8.584 

8.069 

8.285 

8.580 

8.307 

log p —9—2 log s 

0.569 

0.416 

0.931 

0. 715 

0. 420 

0.693 

p ofh2—h t 

3.71 

2.61 

8.53 

5.19 

2.63 

4.93 

Station 1 

Gordon 

Gordon 

Gordon 

Child 

Child 

Rattle 

Station 2 

Red Moun- 

Rattle 

Child 

Rattle 

Red Moun- 

Red Moun- 


tain 




tain 

tain 


O t ff 

O / ft 

O t ff 

O t ft 

0 r rr 

O / ft 

Ci 

90 03 28 

90 31 05 

• 91 56 00 

87 56 00 

88 33 47 

89 07 26 

C2 

90 10 44 

89 38 48 

88 11 27 

92 09 03 

91 36 13 

90 57 56 

C2-C1 

+ 07 16 

- 52 17 

- 3 44 33 

+ 4 13 03 

+ 3 02 26 

+ 1 50 30 

KC2-Cl) 

+ 03 38 

- 26 08.5 

- 1 52 16.5 

+ 2 06 31.5 

+ 1 31 13 

+ 55 15 

tan KC 2 -C 1 ) 

7.02404 

7.88106 

8.51416 

8.56610 

8. 42390 

8.20610 

logs 

4.48780 

4.31274 

4.23223 

4.03839 

4.34662 

4.06606 

log s tan KC 2 -C 1 ) 

1.51190 

2.19380 

2. 74639 

2.60449 

2.77052 

2.27216 

s tan i(C 2 —Ci) 

+32.50 

-156.24 

-557.69 

+402.24 

+589.55 

+ 187.14 

Second term 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

Third term 

0.01 

0.03 

0.09 

0.06 

0.09 

0.04 

h 2 -hi 

+32.51 

-156.27 

-557.78 

+402.30 

+589.64 

+187.18 

2 log s 

8.976 

8.625 

8.464 

8.077 

8.693 

8 132 

log p= 9—2 log s 

0.024 

0.375 

0.536 

0.923 

0.307 

0.868 

p of h 2 —hi 

1.06 

2.37 

3.43 

8.3S 

2.03 

7.38 






















APPLICATION OF LEAST SQUARES TO TRIANGULATION. 199 


Computation of elevations from nonreciprocal observations 


Station Occ. 1. 

Pollywog 

Pollywog 

Pollywog 

Elk 

Pack Sad- 

/41a 

Long Ridge 

Station Obs. 2. 

Bosley 

Stack 

Craggy 

Bosley 

<110 

Bosley 

Pack Sad- 







die 

Obj. sighted 








O / // 

o / // 

O f // 

o / // 

O / // 

O / // 

C 

89 11 00 

89 40 00 

88 04 00 

88 40 30 

89 37 38 

91 01 00 

90°-C 

49 00 

20 00 

1 50 00 

1 19 30 

22 22 

-1 01 00 

90 c in secs. 

2940 

1200 

6960 

4770 

1342 

3000 

log ditto 

3.4G835 

3.07918 

3.842(il 

3.07852 

3.12775 

3.50348 

T 

4.08500 

4.08558 

4.08574 

4.0S505 

4.68558 

4.08502 

logs 

4.17103 

4.49539 

4.20117 

4.33291 

4.42745 

4.15591 

log $ cot C 

2.32558 

2.20015 

2.72952 

2.09708 

2.24078 

2.40501 

a and mean <f> 

95 42.2 

141 42.3 

175.9 42.3 

157.0 42.1 

137.8 42.1 

151.8 42.0 

log (0.5—m) 

9.04107 

9.04107 

9.04107 

9.03990 

9.60200 

9.60001 

2 log s 

8.34320 

8.99078 

8.40234 

8.06582 

8.85490 

8.31182 

log (0.5—m) s- 

7.98493 

8.03245 

8.04401 

8.30572 

8.51750 

7.97183 

log p 

0.80535 

6.80439 

G.80370 

0.80399 

0.80440 

0.80409 

log (2d term) 

1.17958 

1.82800 

1.24025 

1.50173 

1.71304 

1.10774 

s cot c 

+211.03 

+ 182.03 

+530. 44 

+497.83 

+ 174.09 

-254.10 

Second term 

15.12 

07.31 

17.39 

31.75 

51.05 

14.71 

Third term 

0.01 

0.01 

0.04 

0.04 

0.01 

0.01 

t—0 

1.51 

1.51 

1.51 

1.45 

1.50 

1.40 

h»—hi 

+228.27 

+250.80 

+555.38 

+531.07 

+227.25 

-237.98 

log p= 9—2 log s 

0.057 

0.009 

0.598 

0.334 

0.145 

0.688 

V 

4.54 

1.02 

3.90 

2.16 

1.40 

4.88 

Station Occ. 1. 

Bald Hill 

Rattle 

Red Moun- 







tain 




Station Obs. 2. 

Red Moun- 

Redding 

Redding 





tain 

Rock 

Rock 




Obj. sighted 


Water level 

Water level 





O / // 

o / // 

o / // 




c 

88 41 22.5 

91 51 30 

92 32 45 




90°-c 

+ 1 18 37.5 

- 1 51 30 

- 2 32 45 




tan 90°-C 

8.35930 

8.51115 

8.04799 




logs 

4.44942 

4.50705 

4.48393 




log s cot C 

2.80878 

3.07880 

3.13192 




a and mean <£ 

21.4 41.0 

31.3 41.5 

48.0 41.4 




log (0.5—m) 

9.02941 

9.04207 

9.02747 




2 log s 

8.8988# 

9.13530 

8.90786 




log (0.5—m) s 2 

8.52825 

8.77737 

8.59533 




log p 

0.80387 

0.80410 

0.80400 




log (2d term) 

1.72438 

1.97327 

1.79007 




s cot C 

+043.84 

-1198.95 

-1354.94 




Second term 

+ 53.01 

+ 94.03 

+ 01.75 




Third term 

0.00 

0.21 

0.27 




t—0 

1.30 

1.45 

- 1.43 




h2—hi 

+098.27 

-1103.20 

-1291.49 




Cor. for reduc- 


+ 0.15 

+ 0.82 




tion to mean 







sea level 







h 2 —hi (corrected) 


-1103.11 

-1290.67 




log p= 9—2 log s 

0.101 

9.805 

0.032 




V 

1.20 

0.73 

1.07 





The adjustment of vertical observations as practiced in the United 
States Coast and Geodetic Survey is made by means of observation 
equations and differs somewhat from the method of conditions. Of 
course condition equations could he employed if it were desired, just 
as triangulation can be adjusted by observation equations. (See the 
adjustment by the Variation of Geographic Coordinates, p. 91 et seq.) 

Elevations for the various stations are assumed somewhat near 
what the final values will be. To these are added x’s to be deter¬ 
mined by the adjustment. (See table of assumed elevations on p. 200.) 
By means of these, observation equations are formed by the com¬ 
parison of the assumed Ti 2 — h t with that determined by computation. 





















200 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

The method of formation is shown below and a tabulated form of all 
of the computation on page 201. 

Fixed elevations 

Bosley. 

Stack. 

Craggy. 

Redding Rock. 

Mean sea level. 


Meters 
1037. 35 
1062. 69 
1368. 31 

| 0.00 


Assumed and adjusted elevations 


Station 

Elevation 

Assumed 
+ correc¬ 
tion 

Adjusted 

Pollywog 

Elk 

Pack Saddle 

High Divide 

Long Ridge 

Bald Hill 

Gordon 

Child 

Rattle 

Red Mountain 

Meters 
811+n 
507+X 2 
817+Xs 
710 +x 4 
1059+Xs 
589+Xe 
1259 -f-X; 
701+Xg 
1103+X9 
1290+xio 

Meters 
811.06 
504.61 
815.74 
708.77 
1055.96 
585.16 
1256.12 
698.19 
1100.46 
1287.70 


FORMATION OF OBSERVATION EQUATIONS 

The observation equations are formed as follows: 

(1) Pollywog, assumed elevation* = 811-faq 
(2) Craggy, fixed elevation = 1368.31 

h 2 — \ (assumed)* =+557.31—^ 

\ — ~h x (observed)* = + 555.38 + v t 

Observed — assumed* = — 1.93 + x t + v t = 0 

— v t — — 1.93 + aq 

p = | of 3.96 = 1.32 

(1) Elk, assumed elevation* = 507+ z 2 

(2) Pollywog, assumed elevation* = 811+3^ 

7i, 2 —(assumed)* = + 304 + 3^ — x 2 

7^ — 7^ (observed)* = +303.75 + v 5 

Observed — assumed* = —0.25 — ic 1 + x 2 +v 6 = 0 

— v 5 = —0.25-x 1 -hx 2 

p = 2.60 

In a similar manner the remaining equations are formed. These 
are usually formed as in the following table. The constant term is 
found in the column “Observed minus assumed,” and the remainder 
of the equation in the column “Symbol.” 


♦Including symbolic correction. 













Table of formation of observation equations 


APPLICATION OF LEAST SQUARES TO TRIANGULATION. 201 


a 


0 2 
0 

8 


ioio»oooooc?i?o'» , ioo55C 
— t'00N(0i(J'rOK5l'Ol'"inNH(6 
Td2!0S2i2;5!i2f'~S2i22 o '- lt ^'- H mo5oo , <j'Niohnc < ju5oocn 
^NOCOINOJNOIVMWHCOOOOOHHOOIOOOOOOO 

'fOioNooH^cowMOHoijdNHdHMNdddriddd 


<N -r 

OHO 


*> + oo 1-4 r-< eor-ioicoo c 4 tht -5 ooo 

I + +1 + 1 I++I I++I + I |+ I +- I 




*0 -O 

fig 

fijs 

< o 


Vo 

a s 

co o 


*T O C3 
•—»<1> L> 

- 0*0 “ 

"*^'3 


© rCi 

*© 


% 

a 

>» 

co 


5 r „-o 

K 2 © 

c g a 
s-i § 

-2S3I 

o « 


•0 g 
© 2 


- a 

0 . 2 j 
£ — •0 i 
©hog I 

© •« 

o 'S ’© 


,0 

u>_ 


O) 

0 

o 

-*-» 

C3 

4-> 

CQ 


CJ 

.2 

v> 

$ 

CO 


O O *3 CO CO t— 


_ M 00 HH 


>oiooocoonio<ncohcowh 


> oo C'l o 
. . . _ _ .jocoocc 

| H Tf o lO H ^ O *0*0 


t > - rH CO O O C vD ! 
*OWD(NWOH(N < 
*0 Cl (N *0 CO CO W < 


)Nfl5(NWHOOa( 

>o^*o^c^r^t-^< 

li-HC0*OC^f-H^cO*O< 


iss 


6C^J © © >,©>-> 
w-0!0. 


© © 

S3S 

•sg 

C«P 

.+dxs 




0 .M 

3 8 

otf 


d4 

’S.bsa.^s - ©3 r 

^QtfKfttf.ssggl.a gg 

«W4 ©©t>> ®P>> \d H © rt bfi—« rj bc;0 '0-0'0l]2'2 , 0_] 

2111=3*^I'gaiSalSgStg’S’a 3 3'g'SS s’S 

OmnMP^W«^W^KWPHK^WW.-!«»OOe£5«oC« 


tobDbO 

o o o 


^'N01C40040C|OOb>NC)OOOWO'PHONOlOC10N^H 

(NONHN^NNddHHMHNtflddHHdddddHo' 

++i++iii+7++ii++1i +1++1++1 + 


JOOONOINNhXHHON© 


)OHH©OMOI!ON 
■|OMiO»COiO» 
lOUJH'fl'HH 


O* 


!>:r^5S t ^P2'J! Q S 0:iC5C, ^' ,, ® l00( ^ 0 c> t ^ t ^ , ^c , o i ocoiocotHo 

KI-OOt'TfOiOcCHNNOCIOOOHHClMOliHHOOOO 

HOHHNooNNNddcidddooiNi'dddcio'dd 

+ + I + + I I I + I ++ I I ++ 1 I +- I + + I + + I + 


L'JWOHiOMHOOONOinNHOOiOOOI'OOMHONO'^ 
CtOlNNHHOOr-IOi-ICC(N4O00O5C0lHt^lO»O<^lO'5 : l<M5OOI 

£88383?! iSS3a955SSS§B w 8S§3aS 

+++++1 + i i +-1 i i ++ i i 17 i i ++-7 i ++ 


H H H'HHHSHfffiHHN H H H H !? (? H 
+ + +++++++++++o+++l + + I 

+ + + + I I + I I II I I I | I I | + I I | ++ | | + 


HOrtOOOOMMNHOVMOHHOOOHOOOOOO 

I I ++• I 1++1++I++I I + + I +- I 1 + 1 I + + 


CltON 

00000*0 00 

*0 *0 »“* ^ r-H t-H 

+++++ i + i i +■ i i i ++-1 i 17 i i +-+7 i ++ 


+++++i+ii+iii++iiiiii++ii++ 


NHHCqO®NMH»M'V«HC?MHa©N©WM^OONOO 

C0CC»i5C^4©X'T'00»O00l0lC0OC^t©0>C0'-IC'5'J < O-fl’O<NC0C0C<5 

+ o + ociooc4 + + aoc<j + cdoi + oiiooo‘ + edoioodc4j> 


a 0 0 

25Shh©o® JJ-2 

3t3C3 2 .h MMM 0 0 0 

' 0 ’ 0 ' 0 >p>' 0 ’ 0 ' 00 j :0 000 

a*5fi5««psb w g g 

■ W/N -* - - ' . *0 'C +-> 


•> Id Jji, Id b£«-j Ti ^ . 

& 030 J >4 m 'S'S © aoto 0 0 022 £ E S'cd’tJ'C.and-^-s-a-s 

33o^Hraa©0 , "OOO®“OOO©®©^i;S^ , s^ 
* * * * * * * * * 


0 

0 * 

a 

o 

© 

© 

4d 


’§>. 

M 
0 
. Pi 

0 a 
.2 8 
2 
>• 


02 

o- 

2 

5 

0 

,0* 

’© u 

© O 

g* 

0 ^ 

S T5 

03 

S § 
X) S 
© 0 

0 ° 

l| 


convenience in 





























202 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


COMPUTATION OF PROBABLE ERROR 

V 2 pv 2 

No. observations-N o. unknown^ 

_ / 0.455 I pv 2 

'\l n 0 Tin 

2^ 2 = 146.394, log = 2. 16552 
No. observations— No. unknowns = 27 —10= 17, colog = 8. 76955 

Constant = 0.455, log = 9. 65801 
log (probable error) 2 = 0. 59308 
log probable error = 0. 29654 
Probable error of unit weight = ± 1.98 * m. 

log (probable error, unit weight) 2 = 0. 59308 
Weight coefficient for Long Ridge = 2.843, log = 0. 45378 

log (probable error) 2 = 0. 13930 
log probable error = 0. 06965 
Probable error for Long Ridge = ±1.17 m. 

FORMATION OF NORMAL EQUATIONS BY DIFFERENTIATION 

The following equations are formed from the table just given: 

— v x =— 1.93±x 1 
—v 2 =— 0.83 ±x x 
— ^ 3 =±1.92±a: 1 
—x 4 = ±0.72±x 2 
—x 5 = — 0.25— x,±x 2 

and so on for the rest of the 27 p’s. 

The function u to be made a minimum is Zp n v n 2 , or 

tt=±1.32(—1.93±Xj) 2 ±0.34(—0.83±x 1 ) 2 ±1.51(±1.92±x 1 ) 2 ±0.72(±0.72±x 2 ) 2 

±2.60(-0.25-x 1 ±x 2 ) 2 ±10.89(-0.73-x 2 ±x 3 ) 2 ±0.47(±6.90±x3) 2 ±2.82(±3.90 
-x 1 ±x 3 ) 2 ±4.54(-3.55-x 2 ±x 4 ) 2 ±4.89(±2.1G-x 3 ±x 4 ) 2 ±8.53(±1.61-x 4 ±x 5 ) 2 
±2.34(-0.12-x 2 ±x 5 ) 2 ±1.63(±4.02-x 3 ±x 5 ) 2 ±3.71(±2.90-x 4 ±x 6 ) 2 ±2.63(- 0.08 
—x 5 ±x 6 ) 2 ±4.93( —1.58—x 6 ±x 7 ) 2 ±2.61(±l.83—x 4 ±x 7 ) 2 ±5.19( ±0.31—x 5 ±x 7 ) 2 
±0.36(-0.67 ±x 10 ) 2 ±0.42(±2.73-x 6 ±x 10 ) 2 ±1.06(-1.51-x 7 ±x 10 ) 2 ±3.43 
(—0.22—x 7 ±x 8 ) 2 ±2.03(±0.64±x 8 —x 10 ) 2 ±0.24(—0.11±x 9 ) 2 ±8.38(—0.30—x 8 ±z 9 ) 2 
±2.37(±0.27—x 7 ±x 9 ) 2 ±7.38(±0.18±x 9 —x 10 ) 2 . 

The function will be rendered a minimum by equating to zero the 
partial differential coefficients with respect to x lf x 2 , etc. By this 
means the following equations are derived: 

±1.32(-1.93±x 1 )±0.34(-0.83±x 1 )±1.51(±1.92±x 1 )-2.60(-0.25-x 1 ±x 2 )-2.82 
(±3.90—x^Xg^O 

±0.72(±0.72±x 2 )±2.60(-0.25-x 1 ±x 2 )-10.89(-0.73-x 2 ±x 3 )-4.54(-3.55-x 2 ±x 4 ) 
—2.34(—0.12 —x 2 ±x 5 )=0 


* This vertical net is not of a high degree of accuracy, it being a small spur of secondary triangulation 
that was executed in some haste with slight attention to vertical observations. It was selected on account 
of its small size. The more accurate work is usually in larger nets. See list of probable errors ranging 
from ±0.23 m. to ±1.83 m. in United States Coast and Geodetic Survey Special Publication No. 13. 








APPLICATION OP LEAST SQUARES TO TRIANGULATION. 203 


+10.89(—0.73—x 2 +a: 3 )+0.47(+6.90+a: 3 )+2.82(+3.90—a^+Xg)—4.89(+2.16—x 3 +:r 4 ) 
—1.63(+4.02—2: 3 +x 5 )=0 

+4.54(—3.55—x 2 +a: 4 )+4.89(+2.16—x 3 +a: 4 )—8.53(+1.61—£ 4 +x 5 )—3.71(+2.90 
—x 4 +a: 6 )—2.61(+1.83—£ 4 +x 7 )=0 

+8.53(+1.61 —£ 4 +a: 5 )+2.34( —0.12 —x 2 +a: 5 )+1.63(+4.02 —2: 3 +2: 5 ) —2.63( —0.08 
-x 5 +a: 6 )-5.19(+0.31-:r 5 +a: 7 )=0 

+3.71(+2.90—x 4 +a: 6 )+2.63(—0.08—x 5 +a: 6 )—4.93(—1.58—x 6 +a: 7 )—0.42(+2.73 

-^e+^io)=0 

+4.93( —1.58—2: 6 +2: 7 )+2.61(+1.83—2: 4 +2: 7 )+5.19(+0.31—2: 5 +2: 7 )—1.06(—1.51 
—x 7 +:e 10 )—3.43(—0.22—x 7 +a: 8 )—2.37(+0.27—a: 7 +:r 9 )=0 
+3.43(—0.22—2: 7 +2: 8 )+2.03(+0.64+2: 8 —2: 10 )—8.38(—0.30—2^+2: 9 )=0 
+0.24(—0.11 +x 9 )+8.38(—0.30—x 8 +:r 9 )+2.37(+0.27— a: 7 +a: 9 )+7.38(+0.18+Xg 

-2:10 )=0 

+0.36(—0.67+2: 1 o)+0.42(+2.37—2: 6 +2: lo )+1.06(—1.51— 2 ^+ 2 : 10 )—2.03(+0.64 
+x 8 —x 10 )—7.38(+0.18+a: 9 —x 10 )=0 

By multiplying and collecting, we obtain the following normals: 


+8.59zi— 2.60x2— 2.82x3. —10.2786=0 

-2.60m-21.09x2-10.89xj- 4.54x 4 - 2.34x5... +24.2159=0 

-2.82i!-10.89X2+20.70X3- 4.89x 4 - 1.63x5. -10.8237=0 

— 4.54x2— 4.89x 3 +24.28x 4 — 8.53x5— 3.71xe— 2.6IX7. —34.8232=0 

- 2.34X2- 1.63X3- 8.53X4+20.32X5- 2.63x 6 - 5.19x 7 . +18.6066=0 

— 8.71X4— 2.63X5+11.69x«- 4.93X7-.0.42xio.+17.1914=0 

— 2.61x4— 5.19x5— 4.93X6+19.59X7- 3.43x g - 2.37x fl — 1.06xio.+ 0.3111=0 

— 3.43X7+13.84X8- 8.38x # - 2.ft3xio. + 3.0586=0 

— 2.37x7- 8.38X8+18.37X9— 7.38iio. — 0.5721=0 

— 0.42x6— 1.06x7— 2.03X8- 7.38x9+11.25xio. — 3.3228=0 


(See the table of normals on p. 204.) 

The normals are most conveniently formed from the table given 
on page 204. The various observation equations are written along 
the horizontal lines in the columns of their respective x’s. The nor¬ 
mals are then formed as in condition equations, except that the 
constant terms must also be multiplied by each column and the sums 
taken for the constant terms in the normals, as may be seen from 
the direct computation of the normals above. • 

After the x’s are determined from the solution of the normals, 
they are added to the assumed elevations, giving the adjusted final 
elevations. The v’s are most easily determined by computing h 2 — \ 
from the adjusted values; if the observed Ji 2 — is subtracted from 
the adjusted value the respective v results. They could, of course, 
be computed by substituting the x’s in the observation equations, 
but this would require more work. 

For a check the 2 pv at any station should equal zero, with the 
possible exception of a small amount due to dropping the decimals 
on the x’s. In the table on page 201, use pv from the first column if 
the x is positive and from the second column if the x is negative. 













204 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28 


Table for formation of normal equations 



V 

N 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

pN 



x’s 

1 

1.32 

-1.93 

+1 










- 2.5476 


1 

+0.0595 

2 

0.34 

-0.83 

+ 1 










- 0.2822 


2 

-2.3925 

3 

1.51 

+ 1.92 

+ 1 










+ 2.8992 


3 

-1.2577 

4 

0. 72 

+0.72 


+ 1 









+ 0.5184 


4 

-1.2304 

5 

2.60 

-0.25 

-1 

+ 1 









- 0.6500 


5 

-3.0402 

6 

10.89 

-0.73 


-1 

+ 1 








- 7.9497 


6 

-3. 8403 

7 

0.47 

+6.90 



+ 1 








+ 3.2430 


7 

-2.8757 

8 

2.82 

+3.90 

-1 


+ 1 








+10.9980 


8 

-2.8107 

9 

4.54 

-3.55 


-1 


+ 1 







-16.1170 


9 

-2.5441 

10 

4.89 

+2.16 



-1 

+ 1 







+ 10.5624 


10 

-2.2951 

11 

8.53 

+1.61 




-1 

+ 1 






+ 13.7333 




12 

2.34 

-0.12 


-1 



+ 1 






- 0.2S08 



13 

1.63 

+4.02 



-1 


+1 






+ 6.5526 



14 

3.71 

+2.90 




-1 


+1 





+ 10.7590 



15 

2.63 

-0.08 





-1 

+ 1 





- 0.2104 



16 

4.93 

-1.58 






-1 

+ 1 




- 7. 7894 



17 

2.61 

+ 1.83 




-1 



+ 1 




+ 4.7763 



18 

5.19 

+0.31 





-1 


+ 1 




+ 1.6089 



19 

0.36 

-0.67 










+ 1 

- 0.2412 



20 

0. 42 

+2. 73 






-1 




+ 1 

+ 1.1466 



21 

1.06 

-1.51 







-1 



+ 1 

- 1.6006 



22 

3.43 

-0.22 







-1 

+ 1 



- 0. 7546 



23 

2.03 

+0.64 








+ 1 


-1 

+ 1.2992 



24 

0.24 

-0.11 









+ 1 


- 0.0264 



25 

8.38 

-0.30 








-1 

+ 1 


- 2.5140 



26 

2.37 

+0.27 







-1 


+ 1 


+ 0.6399 



27 

7.38 

+0.18 









+ 1 

-1 

+ 1.3284 




Normal equations 



1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

V 

2 

1 

+8.59 

- 2.60 

- 2.82 








-10.2786 

- 7.1086 

2 


+21.09 

-10.89 

- 4.54 

- 2.34 






+24.2159 

+24.9359 

3 



+20. 70 

- 4.89 

- 1.63 






-10.8237 

-10.3537 

4 




+24.28 

- 8.53 

- 3.71 

- 2.61 




-34.8232 

-34.8232 

5 





+20.32 

- 2.63 

- 5.19 




+ 18.6066 

+ 18.6066 

6 






+ 11.69 

- 4.93 



- 0.42 

+17.1914 

+17.1914 

7 







+ 19.59 

- 3.43 

- 2.37 

- 1.06 

+ 0.3111 

+ 0.3111 

8 








+ 13.84 

- 8.38 

- 2.03 

+ 3.0586 

+ 3.0586 

9 









+ 18.37 

- 7.38 

- 0.5721 

- 0.3321 

10 










+ 11.25 

- 3.3228 

- 2.9628 


Solution of normal equations 


1 

2 

3 

4 

5 

6 

7 

V 

2 

+8.59 

- 2.60 

- 2.82 





-10.2786 

- 7.1086 

X\ 

+ 0.30268 

+ 0.32829 





+ 1.19658 

+ 0.82754 


+21.09 

-10.89 

- 4.54 

- 2.34 



+24.2159 

+24.9359 

1 

- 0.7870 

- 0.8536 




• • 

- 3.1111 

- 2.1516 


+20.3030 

-11.7436 

- 4.54 

- 2.34 



+21.1048 

+22.7842 


X 2 

+ 0.57842 

+ 0.22361 

+ 0.11525 



- 1.03949 

- 1.12221 



+20. 70 

- 4.89 

- 1.63 



-10.8237 

-10.3537 


1 

- 0.9258 





- 3.3744 

- 2.3337 


2 

- 6.7927 

- 2.6260 

- 1.3535 



+12.2074 

+13.1788 



+12.9815 

- 7.5160 

- 2.9835 



- 1.9907 

+ 0.4913 



x 3 

+ 0.57898 

+ 0.22983 



+ 0.15335 

- 0.03785 




+24.28 

- 8.53 

-3.71 

-2.61 

-34.8232 

-34.8232 



2 

- 1.0152 

- 0.5232 



+ 4.7192 

+ 5.0948 



3 

- 4.3516 

- 1.7274 



- 1.1526 

+ 0.2845 




+18.9132 

-10.7806 

-3.71 

-2.61 

-31.2566 

-29.4440 




Xi 

+ 0.570004 

+0.196159 

+0.137999 

+ 1.652634 

+ 1.556796 
























































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 205 


Solution of normal equations —Continued 


9 

8 

10 

7 

6 

5 



+18.37 

- 8.38 

- 7.38 

- 2.37 


* 

- 0.5721 

- 0.3321 

Xg 

+ 0.45G18 

+ 0.40174 

+ 0.129015 



+ 0.03114 

+ 0.01808 


+13.84 

- 2.03 

- 3.43 



+ 3.0586 

+ 3.0586 

9 

- 3.8228 

- 3.3666 

- 1.0811 



- 0.2610 

- 0.1515 


+10.0172 

- 5.3966 

- 4.5111 



+ 2.7 16 

+ 2.9071 


X 8 

+ 0.53873 

+ 0.45034 



- 0.27928 

- 0.29021 



+ 11.25 

- 1.06 

- 0.42 


- 3.3228 

- 2.9628 


9 

- 2.9648 

- 0.9521 



- 0.2298 

- 0.1334 


8 

- 2.9073 

- 2.4303 



+ 1.5072 

+ 1.5661 



+ 5.3779 

- 4.4424 

- 0.42 


- 2.0454 

- 1.5299 



XlO 

+ 0.82605 

+ 0.07810 


+ 0.38033 

+ 0.28448 




+19.59 

- 4.93 

- 5.19 

+ 0.3111 

+ 0.3111 



4 

- 0.3602 

- 0.5120 

- 1.4877 

- 4.3134 

- 4.0632 



9 

- 0.3058 



- 0.0738 

— 0.0428 



8 

- 2.0315 



+ 1.2599 

+ 1.3092 



10 

- 3.6696 

- 0.3469 


- 1.6896 

- 1.2638 




+13.2229 

- 5.7889 

- 6.6777 

- 4.5058 

- 3.7495 




xr 

+ 0.43779 

+ 0.50501 

+ 0.34076 

+ 0.28356 





+11.69 

- 2.63 

+17.1914 

+17.1914 




4 

- 0. 7277 

- 2.1148 

- 6.1313 

- 5.7757 




10 

- 0.0328 


- 0.1597 

- 0.1195 




7 

- 2.5343 

- 2.9234 

- 1.9726 

- 1.6415 





+ 8.3952 

— 7.6681 

+ 8.9279 

+ 9.6550 





. x 6 

+ 0.91339 

- 1.06345 

- 1.15006 






+20.32 

+18.6066 

+18.6066 

i 




2 

- 0.2697 

+ 2.4323 

+ 2.6259 





3 

- 0.6857 

- 0.4575 

+ 0.1129 





4 

- 6.1450 

-17.8164 

-16. 7832 





7 

- 3.3723 

- 2.2755 

- 1.8935 





6 

- 7.0040 

+ 8.1547 

+ 8.8188 






+ 2.8433 

+ 8.6442 

+11.4875 






X5 

- 3.04020 

- 4.04020 


Back solution 


5 

6 

7 

10 

8 

9 

4 

3 

2 

1 

-3.0402 

-1.0634 

-2.7769 

+0.3408 

-1.5353 

-1.6812 

+0.3803 

-0.2999 

-2.3755 

-0.2793 

-1.2950 

-1.2364 

+0.0311 
-0.3710 
-0.9220 
-1.2822 

+1.6526 
-1.7329 
-0.7533 
-0.3968 

+0.1534 
-0.6987 
-0.7124 

-1.0395 
-0.3504 
-0.2751 
-0.7275 

+1.1966 
-0.4129 
-0. 7242 

-3.0402 

-3.8403 

-2.8757 

-2.2951 

-2.8107 

-1.2577 

+0.0595 

-2.5441 

-1.2304 

-2.3925 


DEVELOPMENT OF FORMULAS FOR TRIGONOMETRIC LEVELING 

GENERAL STATEMENT 

The formulas used on pages 198 and 199 in the computation of ver¬ 
tical observations were found to be lacking in some of the quantities 
that were appreciable when the lines were very long and high. Accord¬ 
ingly, a new derivation is now given that takes into account some of 
these quantities. As a result, the formulas derived in this develop¬ 
ment differ slightly from those used in the computation cited above, 
but they ought to give practically the same result in computing over 
lines of such length as occur therein. 







































206 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

The following derivation of the formulas for trigonometric leveling 
is based on certain approximate assumptions which fall under four 
general heads : 

1. Geometric approximations. —The verticals at the two points 
(P t the point occupied and P 2 the point sighted on) are treated as if 
they lay in one plane and the intersection of this plane with the 
ellipsoid that represents the surface of the earth is treated as the arc 
of a circle whose radius is the mean radius of curvature of a vertical 
section through P t and P 2 . Helmert (in his Hohere Geodasie, Yol. I, 
p. 520, and Vol. II, p. 563) investigates the error arising from these 
assumptions and finds it to be about 1/40 meter at a maximum when 
the distance P t P 2 is about 100 kilometers. 

2. Geodetic approximations. —The difference between the geodetic 
zenith and the astronomic zenith, i. e., the deflection of the plumb 
line, is ignored. If these deflections are known, corrections may be 
applied to the measured zenith distances (which, of course, are referred 
to the astronomical zenith) to reduce them to the geodetic zenith. 
Furthermore, the elevations obtained by trigonometric leveling 
between two points are referred to an assumed ellipsoid, while spirit 
leveling gives elevations referred to the geoid, so that the distances 
between geoid and ellipsoid must be known to make the two kinds of 
leveling comparable. If trigonometric leveling could be carried out 
with great precision, its use in connection with spirit leveling would 
give just this information as to the distance of the ellipsoid from the 
geoid. The change in the distance from geoid to ellipsoid occurring 
between P x and P 2 may be found from the deflections of the vertical 
at those points, provided it is assumed that the deflections vary 
uniformly between P t and P 2 , an assumption which may be con¬ 
siderably in error. 

3. Optical approximations. —The path of the ray of light between 
P x and P 2 is assumed to be the arc of a circle in a vertical plane 
through P x and P 2 . The angle between the chord P x P 2 and the 
tangent to the circle at either point is the refraction in zenith distance 
and it is evidently implied that this refraction is equal at P x and P 2 . 
If we call O (see figure 9) the center of the circle referred to in approxi¬ 
mation 1, and call the angle PfiP 2 = 0, the refraction in zenith distance 
of the angle TPJP 2 (= Z TPJP^ is written as mO and m is termed the 
coefficient of refraction. The course of a ray of light through the 
atmosphere depends on the variations in pressure, temperature and 
humidity of the medium through which it passes and may be far from 
circular. Our lack of knowledge of the conditions which govern the 
refraction is the greatest obstacle to precision in trigonometric 
leveling. 


APPLICATION OF LEAST SQUARES TO TRIANGULATION. 207 

4. Algebraic approximations.— After the approximations mentioned 
above have been made, there is the further approximation arising 
from the dropping of small terms after an expansion in series. In 
the following developments it will be seen that only extremely small 
terms are dropped, and that in cases arising in practice their effect 
even on the sixth place of logarithms is unimportant, while in fact 
logarithms of only five places are commonly used for this sort of com¬ 
putation. The accuracy of the developments is confirmed by the 
numerical agreement between the approximate and the exact for¬ 
mulas in the examples given. (Exact is used in the sense of dispensing 
with the use of series. The formula is inexact, owing to the first three 
sets of approximations.) The examples represent rather extreme 
cases of those arising in practice, and other numerical examples of 
extreme cases give a similar agreement. 

DEVELOPMENT OF THE FORMULAS 

Figure 9 represents the vertical plane of approximation 1 common 
to P x and P 2 , being in fact the plane parallel to both verticals (see 
Helmert, Iiohere [Geodasie, 

Vol. I, p. 519) on which the 
several points afe projected. 

The measured zenith dis¬ 
tances are assumed equal to 

Z I W = Cl 
and Z F 2 P 2 r = C 2 - 

The measurements are not 
made exactly in this plane, 
but the error, which is part of 
that involved in approxima¬ 
tion 1, is negligible. 

The refraction in zenith dis¬ 
tance is, according to approxi¬ 
mation 3, 

JC=ATP 1 P 2 -=ZTP 2 P 1 =mO. 

jS 1 and S 2 are points on the 
earth’s surface in the verticals 
of P ! and P 2 , so that the re¬ 
spective elevations of P x and P 2 above the surface are 

7q = S 1 P 1 

and Ji 2 = S 2 P 2 . 

The mean radius of curvature p of approximation 1 is given by 

p= os t =os 2 . 



91865°—15-14 




208 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


If s denotes the distance P X P 2 measured along the arc and if 6 be 
expressed in radians, 

s = p0 

or if 6 be in seconds, 

0 " = — J-r 7 >. 
p sin 1" 

There are two cases to he considered according as to whether 
both or only one of the zenith distances have been measured. 

Case 1 . Reciprocal zenith distances 

In the triangle P x OP 2 

/.PJPfi = 180° - Cl - = 180 ° - Cl 

/.PJPfi = 180° - c 2 ~ K = 180° - C 2 - md 
also OP 1 = p + h 1 

and OP 2 = p + h 2 . 

Therefore by the law of sines 

p + h t = sin fe-f -mQ) 
p + h 2 sin 

Treating this as a proportion and taking by division, 

{p + h 2 ) - (p + h x ) _ sin fc+mfl) - sin fe+mfl) 
p + h x sin (£ 2 +md) 


or 


^2 ^1 


: (p+KJ sin cos + md ^ 


sin (C 2 +md) 

Since the sum of the angles of a triangle is 180°, 

180° — — m#+180° — £ 2 —m# + #= 180° 

which gives 

C 1 + C 2 


also 


2 ? +m0 = 9O°+| 


^me^+m.8 + fc£= 90 °4+^ 


whence (A) becomes 


2(p + h 1 ) sin ) sin ~ 


hn h 


6 


■(SS) 


(A) 


( 1 ) 


The quantity 2 (p + h x ) sin ^ has a simple geometrical interpretation 
In the figure make OL 2 = OP 1 and draw OM 1 P X L 2 . Then 

P x M=L 2 M = OP x sin PfiM = (p + h x ) sin 

















APPLICATION OF LEAST SQUARES TO TRIANGULATION. 209 


Q 

Then 2(p + \) sin ^ is the chord P X L 2 or the chord S X S 2 increased to 

allow for the elevation of P x above the earth s surface. In fact, the 
relation (1) might have been obtained by applying the'law of sines 
directly to the trianglo P X P 2 L 2 , which makes it evident why P t L 2 
appears. 

For convenient computation* (1) may be transformed as follows: 
By the sine series 


• • •) 


The remaining factors of the right-hand side of (1) may be written, 


Sm (^) __ sii <S-) _ 

C0M 2 < ' 1 + 0 cos cos f ~ sin 2 sin f 



\ 0 
) sec — 

V 2 

/ 2 

1 — tan ~ tan ( 

) 

= tan(^ 2 Cl )( 



(3) 


Q 

The last transformation comes by expanding sec ^ in powers of 0 

0 @ ^ ^ 
and noting that tan ^ ^ near bb and that the product ~ tan i s 

small, so that, 


1 — 2 ^ an 


(^) 


= 1+2 tan 


(^) 


very nearly. 


By combining (2) and (3) and using 0 = -, equation (1) becomes 

K~ K = « (l +~) tan [l + Y p tan [l + j^] 

or h 2 — \ = s tan ABC 


(4) 


* See also note 1, p. 219. 





















210 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


where A = 1 + - 1 = correction for elevation of station whose elevation 
^ is known, 


B — 1 4- s- tan 

2p 


= correction for approximate difference 
^ ^ / of elevation, 

s 2 

C= 1 = correction for distance. 

12 ( 0 2 

The logarithms of A, B, and C are given in the tables on pages 

218 and 219 with the respective arguments h x , log 

and log s. The tables show the limiting values of the respective 
arguments for which logarithms of A, B, and C become 1, 2, 3, etc., 
units of the fifth place of decimals. 

Equation (4) may be compared with the expression more commonly 
given for h 2 — , 


K - \ = s tan [ 


2 p I2p\ 


(5) 


With the tables here given (5) will probably he found slightly 
more convenient for logarithmic computation than (4). The two 
forms are equally accurate. 

* 

Case II. Only one zenith distance (^) observed 


Where two zenith distances are known, the formula, either (4) or 
(5), does not involve the coefficient of refraction (m) explicitly. Where 
only one zenith distance is known, a value of m must be assumed 
from the best sources of information available. 

In the triangle P 1 L 2 P 2 

IP, Z 2 P 2 = 90°+|=Z V,P,L 2 


ZP 2 P, L = Z V , P, L- z V i P, P 2 

= 90 o +|-(C 1 + ^C) = 90 °-C+(i-»i)^ 


For the third angle we find, by subtracting the sum of the other 
angles from 180° 

/_P X P 2 L = £ i — (1 —m)0. 

By the law of sines 

L 2 P 2 _ sin P 2 P x L 2 
P X L~ sin P x P 2 L 2 


or 


h 2 h x — P x L 2 


cos [Ci — (t—m)fl] 

sin — (1 —md)]' 


( 6 ) 


The chord P X L 2 = chord S x S 2 X - = chord having the 

meaning previously given; chord S x S 2 = &yc s very nearly; or, if 
greater precision is desired, P X L 2 = sAR, where R is the reduction 
factor from arc to chord. 









APPLICATION OF LEAST SQUARES TO TRIANGULATION. 211 


The logarithm of the reduction factor from arc to sine is given in 
the Coast and Geodetic Survey Special Publication No. 8 (Formula) and 
Tables for the Computation of Geodetic Positions), page 17. The 
logarithm of the reduction to chord is very nearly one-fourth of the 
reduction from arc to sine. Granting approximations 1, 2, and 3, 
equation (6) may be rewritten as the .so-called exact formula in the 
following form: 


Tt 2 ^ — sdl? - 


sin 


co s (ci-(i—m)^- 1/7 ) 

[g — m ) p sIh i" - p sin 1"J 


(7) 


Sin 1" is introduced to convert the angle from radians to seconds of 
arc. A and R have the meanings previously indicated. The quan¬ 
tity (J—m) appears in the computation of the refraction from recip¬ 
rocal zenith distances on the Coast and Geodetic Survey forms. A 
mean of the determinations of (£— m) from the reciprocal zenith dis¬ 
tances should be used in computing the nonreciprocal observations. 

s 

Having found the angle — (J — m) ^ sm yi f° r the numerator, the com¬ 


puter should subtract ^ ^ ; in T /7 ^ rom ^ ^o the angle for the 

denominator. The angle in the denominator need not be carried 
out very accurately, as it is always near 90° where the sine varies 
slowly. 

The former Coast and Geodetic Survey formula was 


cot Ci+ li 

21 ^ p p 


( 8 ) 


It is obtained from (6) or (7) by expanding in series and dropping 
certain small quantities. On some of the longer lines the quantities 
dropped are appreciable in computations with five-place logarithms. 
The development hereafter given will show that the general form of 
(8) may be retained by the introduction of correction-factors D t and 
Z> 2 , which are nearly unity, and by the further factor A , the correc¬ 
tion-factor for elevation of the occupied station. The full formula 
will then be, 


Ji 2 — \=A D 1 s cot £ + 


(i— m) AD 2 s? [ 

P 


(l—m) s 2 cot 2 
P 


This form may be obtained from (6) as follows: 
As before 

n 

P 1 L 2 = 2(p + 7i 1 ) sin - 
or expanding by the sine series 



■> 


(9) 












212 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


The factor cos fo — (J— m) 0] in (6) may be written 

cos [Ci -G-mj 0] = cos Ci cos [Q-m) tf] + sin Ci sin [(J-m) 0] 
Since (£ —m) <9 is a small quantity, the series forms for its sine and 
cosine may be used, giving 


cos[£,-(£-m)0] = coss/l-(£-m) 2 ^ + • • -J 

+ sin jli-m) 0-(i-m) 3 ^ + 



( 10 ) 


The third factor on the right-hand side of (6), namely, 

——p--\—Tp; = cosec [cTi — (1 —m)0] 

sm — (1 — m) 6] U1 v 

may be expanded in powers of (1 —m)d by Taylor’s theorem. 


/(Ci) = cosec Ci 
.f(Ci)=-cot Ci cosec Ci 
/"(Ci) = cosec Ci (1 + 2 cot 2 Ci) 

/'"(Ci) = — 6 cosec 3 Ci cot Ci + cot Ci cosec Ci* 

This gives, 


. - r - - jz. -v-^ = cosec Ci + cosec Ci cot r v (l—m) 6 

sm [Ci — (1— m) 6] - 1 ^ ^ 

0 2 

:i (1+2 cot 2 Ci)(l-m) 2 2 
+ cosec Ci cot Ci (6 


+ cosec Ci 

6 s 

cosec 2 Ci — 1) (1 — m) 3 -g- + 


( 11 ) 


The expressions (9), (10), and (11) for the factors on the right-hand 
side of (6) are now to be multiplied together. 

In cases that actually occur, 0 and cot Ci are small quantities of 
about the same order of magnitude. If we call cot Ci a quantity of 
the first order, it is evident that cosec Ci differs from unity by a quan¬ 
tity of the second order. In forming the product from (9), (10), and 
(11) it is seen that the product is of the second order, and will more¬ 
over contain only terms of even order, so that if terms of the fourth 
order are retained the error will be of the sixth order, or the propor¬ 
tional error (the error as compared with the quantity itself) will be 

of the fourth order or of the order of part of the difference of eleva¬ 
tion, if we suppose a quantity of the first order may be as large as i > a 

liberal allowance. The error, then, of the omitted terms should not 
affect the fifth place of logarithms and probably not the sixth. It 
will be seen that the expansions (9), (10), and (11) have been carried 
put sufficiently far for the purpose in hand, and if these expressions 




APPLICATION OF LEAST SQUARES TO TRIANGULATION. 213 

be multiplied together, retaining in the product no terms of higher 
order than the fourth, the result may bo written: 


^ 2-^1 = (p + \) | cot Ci[l + 


+ (i-m) 0 2 1 + 


5 — 10 m + 4 m 2 
~ 12 ~ 


6 (1 —m) 2 — 1 
6 

0 2 ]+(l-m) 


0 2 cot 2 


Since 0 = -> we may write 


=AD 1 s cot 0 + .^Aa-w)g + (l-m)^cot a C 1 
.9 . 9 


where D x = 1 + 
D 2 = 1 + 


6 (1 — m ) 2 — 1 

6 ‘ p 2 

5 — 10m + 4 m 2 


( 12 ) 

(13) 

(13a) 


12 p 2 

. The factor A has been omitted from the last term as being unneces¬ 
sary, the latter being small and A near unity. D x and D 2 are also 
near unity. Their logarithms are tabulated in the same manner as 
the other quantities, the tables showing the limiting values of the 
argument between which log D x or log D 2 may be taken as 1, 2, 3, 
etc., units of the fifth decimal. 

It may be noted that in some European surveys the term 
cot 2 t* 

(1 — m)-— is dropped and the formula for difference of eleva- 

p 

tion written as 


h 2 — li t =s cot Ci + (i—m)- 
P 


(14) 


The dropped terms or factors all represent quantities of the fourth 


order in our expansion. The term 


(1 —m)s 2 cot 2 


is, however, the 


largest of such quantities as a rule, and might be noticeable where 
D x and D 2 would not be. 

Probably for short lines and small differences of elevation the most 
convenient formula would be 

s 2 

\ — As cot^H- A(i— m)- (15) 

and for other lines formula (7). 











214 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 

EXAMPLES 

The data for the following examples, which illustrate the use of 
the formulas, come from The Transcontinental Triangulation, Special 
Publication No. 4, page 273, et. seq. 



At Snow 
Mountain 
West 

At Ross 
Mountain 

c 

<6 

a 

Approx, eley. 

0 / // 

91 13 39.1 
39 22 38 

18 56 18 
2146 meters 

or// 

89 34 04.8 

38 30 20 

197 49 29 

672 meters 


log 5=5.007341. 


For mean a and <f> on the Clarke spheroid of 1866, log p = 6.80369. 

Example 1. Difference of elevation for reciprocal zenith distances, 
assuming Snow Mountain West as the known elevation from for¬ 
mula (5). 

Example 2. Same data as Example 1 worked by formula (4). 

Example 3. Assuming Ross Mountain as known elevation, solve 

by (4). 

Example 4. With refraction from reciprocal zenith distances, but 
with only zenith distance at Snow Mountain West appearing ex¬ 
plicitly, find difference of elevation by (7). 

Example 5. Same data as Example 4, worked by (13). 

Example 6. Like example 4, except zenith distance at Ross Moun¬ 
tain is used. 

Example 7. Like example 5, except zenith distance at Ross Moun¬ 
tain is used. 

The agreement of the differences of elevation as computed by the 
various combinations of data and formulas will give an idea of the 
accuracy of the latter. 






APPLICATION OF LEAST SQUARES TO TRIANGULATION. 215 


Example 1 


Example 2 

Example 3 

Station 1 

/Snow Mountain 
\ West 




Station 2 

Ross Mountain 





Off/ 




Ci 

C2 

C 2 -C 1 

KCs-Ci) 
i(C 2 —0) in secs, 
log ditto 

T 

log s 

log s tan KCi-Ci) 
s tan A(Ca—Ci) 
Second term* 
Third term* 

91 13 39.1 

89 34 04.8 

- 1 39 34.3 

- 49 47.15 

- 2987.15 
3.475257n. 
4.685605 
5.007341 

3.108203W 
-1473.00 

- .03 

- .33 

logs tan3(£n—Ci) 
log A 
log B 
logC 

3.168203?? 

+146 
- 50 

+ 11 

3.168203 
+46 
+50 
+ 11 

hi—hi 

-1473.36 

log (hi— /?i) 

hi—hi 

3.1083107? 
-1473.36 

3.168310 
+1473.36 

M 

hi 

2145.66 
072.30 

/ n 

C 1 +C 2 -ISO 0 

C 1 +C 2 —ISO in secs, 
log ditto 
log p 

47 43.9 
2863.9 
3.456958 
6.803690 





• 4.992659 




log4.38454 

4.384545 




log (0.5— m) 

(0.5 -m) 

9.637852 

0.43436 





Example 4 

Example 6 

logs. 

5.007341 

(Same as ex- 

colog p 

3.190310 

ample 4) 

colog sin 1" 

5.314425 


log 0 f 

3.518076 


log (0.5—77?) 

9.637852 


log (0.5— in)0 

3.155928 


(0.5-77?)!? 

1431. "95 


Cl 

91° 13' 39. "1 

89° 34' 04. "8 

(0.5-77?)!? 

23 51. 95 

23 51. 95 

Cl—(0.5— 77l)<? 

90 49 47. 15 

89 10 12. 85 

d 

2 

27 28 

27 28 

Ci—(1 m)d 

90 22 19 

88 42 45 

logs 

5.007341 

5.007341 

log A 

+ 146 

+ 46 

log R 

- 5 

- 5 

log cos [Ci—(0.5—77?)0] 

8.1608177? 

8.160817 

colog sin [Ci—(1— 777 ) 0 ] 

+ 9 

+ 110 

log (hi—hi) 

3.1683087? 

3.168309 

hi—hi 

-1473.36 

+ 1473.36 


* Second and third terms in example 1 computed by aid of table in General Instructions for Field 
"Work, Coast and Geodetic Survey, pp. 36-37 (edition of 1908). 

, as in the text. 


X 


t d is used for 


p sin 1" 






































216 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


Example 5 

log s 
log A 
log-Di 
cot Cl 


5.0073-11 
+ 146 
+ 79 

8.3309757 X 

log (0.5—771) 
log s- 
colog p 
log A 

9.637852 
10.014682 
3.196310 
146 

log s 2 
colog p 
cot, 2 Ci 
log (1-777) 

10.015 

3.196 

6.662 

9.971 

log first 


3.33854171 

log n 2 

39 

log third 

9.844 

First 

Second 

Third 


-2180.424 
+ 706.365 
+ .698 

log second 

2.849029 



Jl2 -^1 


-1473.361 





Example 7 

logs 
log A 
log Di 
cot Ci 


5.007341 
+46 
+ 79 

7.877369 

log (0.5—777) 
log S 2 
colog p 
log A 

9.637852 
10.014682 

3.196310 
+46 

log S 2 
colog p 
cot 2 Ci 
log (1-777) 

10.015 

3.196 
5.755 
9.971 

log first 


2.884835 

log n 2 

+39 

log third 

8.937 

First 

Second 

Third 


+ 767.070 
+ 706.202 
+ .087 

log second 

2.848929 



hz—hi 


+1473.359 






RECAPITULATION OF FORMULAS 
(Numbered as in foregoing discussion) 

Case I. Reciprocal observations 
Former Coast and Geodetic Survey form, 


7 ?2 — s tan 


(WD 


4- 7i 2 


2 p ' 12 p- 


] 


(5) 


Reference: Page 210 and General Instructions for Field Work Coast and Geodetic 
Survey, pages 34-37 (edition of 1908). 

Logarithmic form, 

h 2 -\ = s tan ^ ABC. (4) 

Reference: Page 209 and tables. 

Case II. Nonreciprocal observations 

Former Coast and Geodetic Survey form, 

7 -i , „ . 0.5—m , , (1 —m) s 2 cot 2 r. 

ti 2 -\ = s cot Ci +-s 2 + - - - - - • (8) 

P P 

Reference: Page 211 and General Instructions for Field Work Coast and Geodetic 
Survey, pages 34-37 (edition of 1908). 

Corrected form, 

AD, s cot c, + AD t s> + (l ~ m)s2 COt2 & . (13) 


Reference: Page 213 and tables. 


P 





































APPLICATION OF LEAST SQUARES TO TRIANGULATION. 217 


“Exact” form, 


7^2 — sA.B 


cos | 

f r f 1 w) s ”1 


L Cl U } P sin 1"J 

sin | 


^ p sin 1" 2 p 

8 1 

sin 1"J 


(7) 


Reference: Page 211 and tables; also Formulae and Tables for Position Computation, 
Coast and Geodetic Survey Special Publication No. 8, for R. 

See also additional note, page 220. 


NOTES ON CONSTRUCTION AND USE OF TABLES 


The tables are constructed with mean values of p and m . 
log ,0 = 6.80444 corresponding to mean radius of curvature in lati¬ 
tude 40° for Clarke’s spheroid of 1866. 

m = 0.06. m varies between 0.05 and 0.10 in the great majority of 
cases. This value near the smaller limit was taken as probably 
nearer the truth for the high lines, in which the correction terms 
tabulated are most likely to appear, than an intermediate value of 
0.07 or 0.08 

A, B, C, D 1 and D 2 are all very near unity. To compute their 
logarithms the approximate expression log (1 +x)=Mx was used, M 
being the modulus of common logarithms =0.43429. 

Formulas for constructing tables: 

^4 = 1+^ = h = -jj 2 log A — 146.78 log A 

log A being in units of fifth place. 

k * B -# 

log [ 5 tan ( ^ 2 2 -^)j = log ^-+log (log B) 

= 7.4677 +log (log B) 


0=1 + 


Di = 1 + 


12 p 2 

6 (1 - to ) 2 


i n Ms 2 

log 12 p 2 


log-log(^/l) + i lo S (log C) 

= 7.5251 + i log (log 0) 


6 


Is 2 , n M 6(1-w) 2 -Is 2 

-3 log D, = M 


Z>o = 1 + 


10 — 20 m + 8 m 2 
24 


6 p 2 

log 5 = 7.0578 + \ log (log A) 

t ir / 5—10m+ 4 m 2 

log D. 


= M (- 


12 


log 8 = 7.2027 + i log flog D 2 ) 


The values of log A, log B, etc., were taken successively at 0.5, 1.5, 
2.5, etc., units of the fifth place, namely, at the point where the value 














218 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 


of log A, log B, as rounded off to 5 decimals would change by one 

in the fifth place. The corresponding values of 7i, log [,.t an (V0] 

and log s were then computed by the formulas above. These values 
are carried out far enough so that the values of log A, log B, etc., may 
be obtained by interpolation to six decimals. In the numerical 
examples here given the values of log A, log B, etc., were computed 
independently for the actual values of p and m. These results as 
used hi the example all agree within a unit of the sixth decimal place 
with those found by interpolating in the tables. 

The unit of length throughout the tables and formulas is the 
meter. 

Tables 


Elevation 
of occupied 
station hi 

log A units 
of fifth place 

Elevation 
of occupied 
station hi 

log A units 
of fifth place 

Meters 

0 

0.0 

Meters 

3009 

20.5 

73 

0.5 

3156 

21.5 

220 

1.5 

, 3303 

22.5 

367 

2.5 

3449 

23.5 

514 

3.5 

3596 

24.5 

661 

4.5 

3743 

25.5 

807 

5.5 

3890 

26.5 

954 

6.5 

4036 

27.5 

1101 

7.5 

4183 

28.5 

1248 

8.5 

4330 

29.5 

1394 

9.5 

4477 

30.5 

1541 

10.5 

4624 

31.5 

1688 

11.5 

4770 

32.5 

1835 

12.5 

4917 

33.5 

1982 

13.5 

5064 

34.5 

2128 

14.5 

5211 

35.5 

2275 

15.5 

5357 

36.5 

2422 

16.5 

5504 

37.5 

2569 

17.5 

5651 

38.5 

2715 

18.5 

5798 

39.5 

2862 

19.5 

5945 

40.5 








APPLICATION OF LEAST SQUARES TO TRIANGULATION. 219 


log A is positive except in the rare case when li { indicates a depres¬ 
sion below mean sea level. 

A is used for both reciprocal and nonreciprocal observations. 


For reciprocal observations onlv (unless 

formula, p. 

—, is used) 


log approx¬ 
imate 
difference 
elevation 
=log stan 

log B 
units of 
5th place 

logs 

log C 

/Cs-Ci\ 



l 2 





0.0 


0.0 

2.167 

0.5 

4.875 

0.5 

2.044 

1.5 

5.113 

1.5 

2.866 

2.5 

5.224 

2.5 

3.011 

3.5 

5.297 

3.5 

3.121 

4.5 

5.352 

4.5 

3.208 

5.5 

5.305 

5.5 

3.281 

6.5 

5.432 

6.5 

3.343 

7.5 

5.463 

7.5 

3.397 

8.5 



3.445 

9.5 



3.4S9 

10.5 



3.528 

11.5 



3.50.5 

12.5 



3.598 

13.5 



3.629 

14.5 



3.658 

15.5 



3.685 

16.5 



3.711 

17.5 



3.735 

18.5 



3. 758 

19.5 



3.779 

20.5 



3.800 

21.5 



3.820 

22.5 



3.839 

23.5 



3.857 

24.5 



3.874 

25.5 




For nonreciprocal observations 


logs 

log Di 
units of 

5 th place 

logs 

log D -2 
units of 
5th place 


0.0 


0.0 

4.407 

0.5 

4.552 

0.5 

4.640 

1.5 

4.791 

1.5 

4.757 

2.5 

4.902 

2.5 

4.830 

3.5 

4.975 

3.5 

4.884 

4.5 

5.029 

4.5 

4.928 

5.5 

5.073 

5.5 

4.964 

6.5 

5.109 

0.5 

4.995 

7.5 

5.140 

7.5 

5.023 

8.5 

5.1G7 

8.5 

5.047 

9.5 

5.192 

9.5 

5.068 

10.5 

5.213 

10.5 

5.088 

11.5 

5.233 

11.5 

5.106 

12.5 

5.251 

12.5 

5.123 

13.5 j 



5.138 

14.5 



5.153 

15.5 



5.167 

16.5 



5.179 

17.5 



5.191 

18.5 



5.203 

19.5 



5.214 

20.5 



5.224 

21.5 



5.234 

22.5 



5.243 

23.5 



5.252 

24.5 



5.261 

25.5 




* Or log s cot £ Ci—(0.5—77i) ^ 1 ; > J for nonreciprocal observations. 


(See note 2, p. 220.) 


log B has the same sign as the approximate difference of elevation. 

log C is always positive. 

log D l and log D 2 are always positive. 


NOTES ON THE DEVELOPMENTS 

Note 1.—The transformation of (1), page 208, may be conducted 
rather more simply than is there given. 

. 6 


h 2 


2 (p + hj sin^ " 2 2 ~)* 


sin 2 


r 

(Ct-C A , °~\ 

cos 1 

V 2 / 2J 


or 


2 (p + hj sin 


h 2 k — 


0 

2 


cos 


A) COS I -sin 2 si 


6 

sin 2 


( 1 ) 












































220 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 
Divide numerator and denominator by cos cos 




^2 ^1 


2 (p + h j) tan k tan 


(s 4 ) 


1 — tan | tan 


(*?:) 

or expanding tan - in series and using 0=-^ 


s tan 


1 + - 


m 


2 P 


= s tan 


which is equation (4). 


(s 6 ) 


ABC, 


Note 2. —The formula for nonreciprocal observations may be put 
in the same form as that for reciprocal observations. 

From the equation on page 208 


£ 2 = 180° — G — 2m d + Q 
90 —£, + (0.5—m) 8 

tan (^2^ J ) = cot fc ~ (0.5-m) d] = cot - (0.5-m) 1 ,/ J 

Substitute in (4) 

h 2 — Ji 1 = s cot [r ± — (0.5 — m) — -*-?» ] ABC 
. -si L '* 1 ^ sm 1 //J 

for nonreciprocal observations analogous to 


l 2 -\ = s tan (%J^) AB<7 

for reciprocal observations, i? should be taken from table with 
argument 

log, cot[ Cl — (0.5 

This is the present Coast and Geodetic Survey formula for non¬ 
reciprocal observations. 


o 






















































I 




* 

























• — 



. 












. 










- 












. 






















































































' 

J* 

’ 
















































































































